MATHEMATICS HIGHER SECONDARY - FIRST YEAR VOLUME - I REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE Untouch ability is a sin Untouchability is a crime Untouchability is inhuman '■">* THOPoW" TAMILNADU TEXTBOOK CORPORATION COLLEGE ROAD, CHENNAI - 600 006 PREFACE This book is designed in accordance with the new guidelines and syllabi - 2003 of the Higher Secondary Mathematics - First Year, Government of Tamilnadu. In the era of knowledge explosion, writing a text book on Mathematics is challenging and promising. Mathematics being one of the most important subjects which not only decides the career of many young students but also enhances their ability of analytical and rational thinking and forms a base for Science and Technology. This book would be of considerable value to the students who would need some additional practice in the concepts taught in the class and the students who aspire for some extra challenge as well. Each chapter opens with an introduction, various definitions, theorems and results. These in turn are followed by solved examples and exercises which have been classified in various types for quick and effective revision. The most important feature of this book is the inclusion of a new chapter namely 'Functions and Graphs'. In this chapter many of the abstract concepts have been clearly explained through concrete examples and diagrams. It is hoped that this book will be an acceptable companion to the teacher and the taught. This book contains more than 500 examples and 1000 exercise problems. It is quite difficult to expect the teacher to do everything. The students are advised to learn by themselves the remaining problems left by the teacher. Since the 'Plus 1' level is considered as the foundation course for higher mathematics, the students must give more attention to each and every result mentioned in this book. The chief features of this book are (i) The subject matter has been presented in a simple and lucid manner so that the students themselves are able to understand the solutions to the solved examples. (ii) Special efforts have been made to give the proof of some standard theorems. (iii) The working rules have been given so that the students themselves try the solution to the problems given in the exercise. (iv) Sketches of the curves have been drawn wherever necessary, facilitating the learner for better understanding of concepts. (v) The problems have been carefully selected and well graded. The list of reference books provided at the end of this book will be of much helpful for further enrichment of various concepts introduced. We welcome suggestions and constructive criticisms from learned teachers and dear students as there is always hope for further improvement. K. SRINIVASAN Chairperson Writing Team SYLLABUS (1) MATRICES AND DETERMINANTS : Matrix Algebra - Definitions, types, operations, algebraic properties. Determinants - Definitions, properties, evaluation, factor method, product of determinants, co-factor determinants. (18 periods) (2) VECTOR ALGEBRA : Definitions, types, addition, subtraction, scalar multiplication, properties, position vector, resolution of a vector in two and three dimensions, direction cosines and direction ratios. (15 periods) (3) ALGEBRA : Partial Fractions - Definitions, linear factors, none of which is repeated, some of which are repeated, quadratic factors (none of which is repeated). Permutations - Principles of counting, concept, permutation of objects not all distinct, permutation when objects can repeat, circular permutations. Combinations, Mathematical induction, Binomial theorem for positive integral /nctex-finding middle and particular terms. (25 periods) (4) SEQUENCE AND SERIES : Definitions, special types of sequences and series, harmonic progression, arithmetic mean, geometric mean, harmonic mean. Binomial theorem for rational number other than positive integer, Binomial series, approximation, summation of Binomial series, Exponential series, Logarithmic series (simple problems) (15 periods) (5) ANALYTICAL GEOMETRY : Locus, straight lines - normal form, parametric form, general form, perpendicular distance from a point, family of straight lines, angle between two straight lines, pair of straight lines. Circle - general equation, parametric form, tangent equation, length of the tangent, condition for tangent. Equation of chord of contact of tangents from a point, family of circles - concetric circles, orthogonal circles. (23 periods) (6) TRIGONOMETRY : Trigonometrical ratios and identities, signs of T-ratios, compound angles A + B, multiple angles 2A, 3A, sub multiple (half) angle A/2, transformation of a product into a sum or difference, conditional identities, trigonometrical equations, properties of triangles, solution of triangles (SSS, SAA and SAS types only), inverse trigonometrical functions. (25 periods) (7) FUNCTIONS AND GRAPHS : Constants, variables, intervals, neighbourhood of a point, Cartesian product, relation. Function - graph of a function, vertical line test. Types of functions - Onto, one-to-one, identity, inverse, composition of functions, sum, difference product, quotient of two functions, constant function, linear function, polynomial function, rational function, exponential function, reciprocal function, absolute value function, greatest integer function, least integer function, signum function, odd and even functions, trigonometrical functions, quadratic functions. Quadratic inequation - Domain and range. (15 periods) (8) DIFFERENTIAL CALCULUS : Limit of a function - Concept, fundamental results, important limits, Continuity of a function - at a point, in an interval, discontinuous function. Concept of Differentiation - derivatives, slope, relation between continuity and differentiation. Differentiation techniques - first principle, standard formulae, product rule, quotient rule, chain rule, inverse functions, method of substitution, parametric functions, implicit function, third order derivatives. (30 periods) (9) INTEGRAL CALCULUS : Concept, integral as anti-derivative, integration of linear functions, properties of integrals. Methods of integration - decomposition method, substitution method, integration by parts. Definite integrals - integration as summation, simple problems. (32 periods) (10) PROBABILITY : Classical definitions, axioms, basic theorems, conditional probability, total probability of an event, Baye's theorem (statement only), simple problems. (12 periods) CONTENTS Page No. Preface Syllabus 1. Matrices and Determinants 1 1.1 Matrix Algebra 1 1.2 Determinants 14 2. Vector Algebra 39 2.1 Introduction 39 41 42 47 58 62 69 69 74 89 98 105 4. Sequence and Series 114 4.1 Introduction 114 4.2 Sequence 115 4.3 Series 117 4.4 Some Special Types of Sequences and their Series 119 4.5 Means of Progressions 121 4.6 Some special types of Series 126 2.2 Types of Vectors 2.3 Operations on Vectors 2.4 Position Vector 2.5 Resolution of a Vector 2.6 Direction Cosines and Direction Ratios 3. Algebra 3.1 Partial Fractions 3.2 Permutations 3.3 Combinations 3.4 Mathematical Induction 3.5 Binomial Theorem 5. Analytical Geometry 133 5.1 Locus 133 135 143 154 160 167 174 179 179 182 197 215 224 232 235 Objective type Questions 243 Answers 255 5.2 Straight Lines 5.3 Family of Straight Lines 5.4 Pair of Straight Lines 5.5 Circle 5.6 Tangent 5.7 Family of Circles 6. Trigonometry 6.1 Introduction 6.2 Trigonometrical Ratios and Identities 6.3 Compound Angles 6.4 Trigonometrical Equations 6.5 Properties of Triangles 6.6 Solutions of Triangles 6.7 Inverse Trigonometrical functions 1. MATRICES AND DETERMINANTS 1.1 Matrix Algebra 1.1.1 Introduction The term 'matrix' was first introduced by Sylvester in 1850. He defined a matrix to be an arrangement of terms. In 1858 Cay ley outlined a matrix algebra defining addition, multiplication, scalar multiplication and inverses. Knowledge of matrix is very useful and important as it has a wider application in almost every field of Mathematics. Economists are using matrices for social accounting, input - output tables and in the study of inter- industry economics. Matrices are also used in the study of communication theory, network analysis in electrical engineering. For example let us consider the marks scored by a student in different subjects and in different terminal examinations. They are exhibited in a tabular form as given below. Tamil English Maths Science Social Science Test 1 70 81 88 83 64 Test 2 68 76 93 81 70 Test 3 80 86 100 98 78 The above statement of marks can also be re-recorded as follows First row Second row 70 81 88 83 64 68 76 93 81 70 Third row L 80 86 100 98 78 First second Third Fourth Fifth Column Column Column Column Column This representation gives the following informations, (i) The elements along the first, second, and third rows represent the test marks of the different subjects, (ii) The elements along the first, second, third, fourth and fifth columns represent the subject marks in the different tests. The purpose of matrices is to provide a kind of mathematical shorthand to help the study of problems represented by the entries. The matrices may represent transformations of co-ordinate spaces or systems of simultaneous linear equations. 1.1.2 Definitions: A matrix is a rectangular array or arrangement of entries or elements displayed in rows and columns put within a square bracket or parenthesis. The entries or elements may be any kind of numbers (real or complex), polynomials or other expressions. Matrices are denoted by the capital letters like A, B, C. . . Here are some examples of Matrices. " 1 4 " First Row 1 -4 2 First row (Rj) 2 5 Second Row " = 6 9 4 Second row (R2) _ 3 6 _ Third Row 3 -2 6 _ Third row (R 3 ) First Second First Second Third Column Column Column Column Column Ci c 2 c 3 Note : In a matrix, rows are counted from top to bottom and the columns are counted from left to right. i.e. (i) The horizontal arrangements are known as rows, (ii) The vertical arrangements are known as columns. To identify an entry or an element of a matrix two suffixes are used. The first suffix denotes the row and the second suffix denotes the column in which the element occurs. From the above example the elements of A are an = 1, ayi = 4, (221 = 2, (222 = 5, 031 = 3 and C132 = 6 Order or size of a matrix The order or size of a matrix is the number of rows and the number of columns that are present in a matrix. In the above examples order of A is 3 x 2, (to be read as 3-by-2) and order of B is 3 x 3, (to be read as 3-by-3). In general a matrix A of order mxn can be represented as follows : d\\ a 12 ••• a \j ■■■ a \n A = an an. — a m \ a m 2 a m j ■lh J column — > i row This can be symbolically written as A = [ay] m x n- The element a ( w belongs to i row and the/ column, i being the row index and j being the column index. The above matrix A is an m x n or m-by-n matrix. The expression m x n is the order or size or dimension of the matrix. Example 1.1: Construct a 3 x 2 matrix whose entries are given by ay = i - 2/ Solution: The general 3x2 matrix is of the form 'an a [2 ~ A = [ay] = a 2l «22 where i = 1, 2, 3 (rows), 7 = 1,2 (columns) _«31 a 32_ It is given that a„- = i - 2/ an = 1 — 2 = — 1 ai2=l-4 = -3 -1 a2l =2-2 = a22 = 2 - 4 = - 2 a31 = 3-2=1 (232 = 3 - 4 = - 1 1.1.3 Types of matrices (1) Row matrix: A matrix having only one row is called a row matrix or a row vector. Examples (i) A = [a y ]i x3=[l -7 4] is a row matrix of order 1x3. (ii) B = [by]i x 2 = [5 8] is a row matrix of order 1x2 (iii) C = [cy]i x 1 = [100] is a row matrix of order lxl (2) Column matrix: A matrix having only one column is called a column matrix or a column vector. .". The required matrix is A = Examples (i) A = [a,y]3 x 1 = (ii) B = [& y ] 2 xi = ' 1 -7 .4. '25" .30. is a column matrix of order 3x1 is a column matrix of order 2x1 (iii) C = [Cy]i x 1 = [68] is a column matrix of order lxl Note : Any matrix of order 1 x 1 can be treated as either a row matrix or a column matrix. (3) Square matrix A square matrix is a matrix in which the number of rows and the number of columns are equal. A matrix of order «x«is also known as a square matrix of order n. In a square matrix A of order n x n, the elements a\i, ai% (233 ... a nn are called principal diagonal or leading diagonal or main diagonal elements. "2 41 is a square matrix of order 2 A = [aij] 2 x 2 B = [bijh x 3 = 6 8. 1 2 3' 4 5 6 7 8 9. is a square matrix of order 3. Note: In general the number of elements in a square matrix of order n is n . We can easily verify this statement from the above two examples. (4) Diagonal Matrix: A square matrix A = [ay] n x n is said to be a diagonal matrix if au = when In a diagonal matrix all the entries except the entries along the main diagonal are zero. For example A = [a^ x 3 = "4 0" 5 _0 6_ is a diagonal matrix. (5) Triangular matrix: A square matrix in which all the entries above the main diagonal are zero is called a lower triangular matrix. If all the entries below the main diagonal are zero, it is called an upper triangular matrix. A = "3 2 7" 5 3 _0 1. is an upper triangular matrix and B = 2 0" 4 1 8 -5 7_ is a lower triangular matrix. (6) Scalar matrix: A square matrix A = [ay] n x n 1S sa id to be scalar matrix if [a if i=j a 'J-[0 if i*j i.e. A scalar matrix is a diagonal matrix in which all the entries along the main diagonal are equal. B = [bij] 3 x 3 A = [aijh x 2 = 5 0' .0 5. ■45 Vs. are examples for scalar matrices. (7) Identity matrix or unit matrix: A square matrix A = [a ( w] n x « is said to be an identity matrix if _ri if t=j a 'J~{0 if i*j i.e. An identity matrix or a unit matrix is a scalar matrix in which entries along the main diagonal are equal to 1. We represent the identity matrix of order n as I„ h = "1 0" "1 0" _0 1_ . 13 = 1 _o 1. are identity matrices. "0 0" "0 0" 5 _0 0_ _0 0_ are examples of zero matrices. (8) Zero matrix or null matrix or void matrix A matrix A = [a,y] m x « is said to be a zero matrix or null matrix if all the entries are zero, and is denoted by O i.e. ay = for all the values of i,j "0 0" [0 0], _0 0_ (9) Equality of Matrices: Two matrices A and B are said to be equal if (i) both the matrices A and B are of the same order or size. (ii) the corresponding entries in both the matrices A and B are equal. i.e. the matrices A = [a,)] m x n an d B = [bjj] p x q are equal if m = p, n = q and ajj = by for every i andj. Example 1.2 : [x y~\ [4 31 If = , - then find the values of x, y, z, w. \_z w_l LI 5J J Solution: Since the two matrices are equal, their corresponding entries are also equal. :.x = A y = 3 z=l w = 5 (10) Transpose of a matrix: The matrix obtained from the given matrix A by interchanging its rows into columns and its columns into rows is called the transpose of A and it is T denoted by A' or A . "4 -3" If A = then A T = 2 1" 5. ,T- Note that if A is of order m x n then A is order nxm. (11) Multiplication of a matrix by a scalar Let A be any matrix. Let k be any non-zero scalar. The matrix kA is obtained by multiplying all the entries of matrix A by the non zero scalar k. i.e. A = yo.ij\ m x n — ^ kA. = yka\j\ m x n This is called scalar multiplication of a matrix. Note: If a matrix A is of order m x n then the matrix kA is also of the same order m x n For example If A = (12) Negative of a matrix: Let A be any matrix. The negative of a matrix A is - A and is obtained by changing the sign of all the entries of matrix A. 1-6- A = i&ij\ m x n —? ~ A = \ — @ij\m x n 1 7 2" then 2A = 2 "17 2" " 2 14 4 ■6 3 9. _-6 3 9_ _- 12 6 18 Let A = then ■A = -cos9 sin9 - sin9 cos9. cos9 sin9 - sin9 cos9_ 1.1.4 Operations on matrices (1) Addition and subtraction Two matrices A and B can be added provided both the matrices are of the same order and their sum A + B is obtained by adding the corresponding entries of both the matrices A and B i.e. A = [aij] m x „ and B = [bij] m x „ then A + B = [ay + bij\ m x „ Similarly A - B = A + (- B) = [a,y] w x „ + [- by] m x „ = \flij ~ Dij\m x n Note: (1) The matrices A + B and A - B have same order equal to the order of AorB. (2) Subtraction is treated as negative addition. (3) The additive inverse of matrix A is - A. i.e. A + (- A) = (- A) + A = O = zero matrix For example, if A = "7 2 " 4 -7" 8 6 andB = 3 1 _9 -6. _-8 5_ then A + B = A-B = A + (-B) = "7 2 " 4 -7" 8 6 + 3 1 = _9 -6. _-8 5_ 7 + 4 8 + 3 9-8 2-7" "11 -5" 6+1 = 11 7 6 + 5_ 1 -1. ~7 2 " "-4 7" 8 6 + -3 -1 = _9 -6. _ 8 -5. 7-4 8-3 9 + 8 and 2 + 7" " 3 9" 6-1 = 5 5 6-5_ .17 -11. (2) Matrix multiplication: Two matrices A and B are said to be conformable for multiplication if the number of columns of the first matrix A is equal to the number of rows of the second matrix B. The product matrix 'AB' is acquired by multiplying every row of matrix A with the corresponding elements of every column of matrix B element-wise and add the results. This procedure is known as row-by-column multiplication rule. Let A be a matrix of order m x n and B be a matrix of order n x p then the product matrix AB will be of order m x p i.e. order of A is m x n, order of B is n x p Aiumber of rows^ /number of columns^ Then the order of AB is m x p = [ rf matrk A )x{ of matrix B ) The following example describes the method of obtaining the product matrix AB Let A = '2 1 4' 7 3 6. B = 2x3 6 4 3' 3 2 5 .7 3 1. 3x3 It is to be noted that the number of columns of the first matrix A is equal to the number of rows of the second matrix B. .". Matrices A and B are conformable, i.e. the product matrix AB can be found. "6 4 3" "2 1 4" AB = 1 3 6_ 3 2 5 _7 3 1. 2 1 4 7 3 6 2 1 4 2 1 4 7 3 6 6 7 3 6 4 3 2 7 3 "(2) (6) + (1) (3) + (4) (7) (2) (4) + (1) (2) + (4) (3) L(7) (6) + (3) (3) + (6) (7) (7) (4) + (3) (2) + (6) (3) "12 + 3 + 28 8 + 2+12 6 + 5 + 4 "1 _42 + 9 + 42 28 + 6+18 21 + 15 + 6. It is to be noticed that order of AB is 2 x 3, which is the number of rows of first matrix A 'by' the number of columns of the second matrix B. Note : (i) If AB = AC, it is not necessarily true that B = C. (i.e.) the equal matrices in the identity cannot be cancelled as in algebra, (ii) AB = O does not necessarily imply A = O or B = O 3 5 1 -I (2) (3) + (1) (5) + (4) (1) (7) (3) + (3) (5) + (6) (1)J "43 22 15 _93 52 42. AB: For example, A = but AB = 1 -1 1 1 "1 1 1 1 * O and B = 1 1" 1 1. *o 0' .0 0. = o (iii) If A is a square matrix then A. A is also a square matrix of the same order. AA is denoted by A . Similarly A A = AAA = A If I is a unit matrix, then I = I 2 = I 3 = . . . = I". 1.1.5 Algebraic properties of matrices: (1) Matrix addition is commutative: If A and B are any two matrices of the same order then A + B = B + A. This property is known as commutative property of matrix addition. (2) Matrix addition is associative: i.e. If A, B and C are any three matrices of the same order thenA+(B + C) = (A+B)+C. This property is known as associative property of matrix addition. (3) Additive identity: Let A be any matrix then A + = + A = A. This property is known as identity property of matrix addition. The zero matrix O is known as the identity element with respect to matrix addition. (4) Additive inverse: Let A be any matrix then A + (- A) = (- A) + A = O. This property is known as inverse property with respect to matrix addition. The negative of matrix A i.e. - A is the inverse of A with respect to matrix addition. (5) In general, matrix multiplication is not commutative i.e. AB * BA (6) Matrix multiplication is associative i.e. A(BC) = (AB)C (7) Matrix multiplication is distributive over addition i.e. (i) A(B + C) = AB + AC (ii) (A + B)C = AC + BC (8) AI = IA = A where I is the unit matrix or identity matrix. This is known as identity property of matrix multiplication. Example 1.3: If A = Prove that "1 8" "1 3" ~-4 6" _4 3_ B = 1 4_ C = . 3 -5_ (i) AB * BA (iii) A(B + C) = AB + AC (ii) A(BC) = (AB)C (iv) AI = IA = A Solution: (i) AB = BA = 1 8 4 3 '1 + 56 .4 + 21 1 3 .7 4 1 + 12 7 + 16 T 3' 7 4. 3 + 32" 12 + 12. T 8' 4 3. 8 + 9 " 56 + 12. (1) (1) + (8) (7) .(4) (1) + (3) (7) '57 35" 25 24_ (1) (1) + (3) (4) .(7) (1) + (4) (4) 13 17" .23 68_ (1) (3) + (8) (4)' (4) (3) + (3) (4). ...(1) (1) (8) + (3) (3)' (7) (8) + (4) (3). • ••(2) From (1) and (2) we have AB * BA (ii) (AB)C = "57 25 35" 24. -4 3 from (1) (57) (- 4) + (35) (3) .(25) (-4) + (24) (3) (57) (6) + (35) (-5) (25) (6) + (24) (-5). (AB)C = BC = BC = A(BC) = -228 + 105 342 - 175 _ - 100 + 72 150 - 120. - 123 167" - 28 30 _ ...(3) 1 3" "-4 6" 7 4_ _ 3 -5_ (l)(-4) + (3)(3) (1) (6) + (3) (- 5)" -4 + 9 6-15 (7) (-4) + (4) (3) (7) (6) + (4) (- 5). _- 28 +12 42 - 20 5-9" -16 22_ 1 8' 4 3. 5 16 -9 22. (1) (5) -(4) (5) "-123 _ -28 From (3) and (4) we have, (AB)C = A(BC) A(BC) = + (8) (-16) (l)(-9) + (8)(22) + (3) (-16) (4) (-9) + (3) (22). 167' 30. 5-128 20-48 ■9+176 36 + 66. •••(4) (iii) B + C = A(B + C) = A(B + C) = AB = AC = AB + AC = "1 3" _7 4_ "1 8" _4 3_ H h "-4 6 _ 3 -5 -3 9" 10 -1. 1-4 3 + 6" "-3 9" 7 + 3 4-5_ _ 10 -1. -3 + 80 9-8" 12 + 30 36- -3. "77 1" .18 33. "57 35" .25 24_ (5) "1 _4 "57 _25 "77 .18 35" 24_ 1" 33. ... from (1) -4 6 3-5. 20 -7 34 9. 24 6 - 40" ': 20 -34" f 9 24 - 15. -i 9_ 57 + 20 35 - 34" 25-7 2 i + 9_ (6) 10 From equations (5) and (6) we have A(B + C) = = AB + AC (iv) Since order of A is 2 x 2, take I = "1 0" _0 1. AI = "1 8" _4 3_ "1 0" _0 1. = "1(1) + 8(0) -4(1) + 3(0) 1(0) + 8(1)" 4(0) + 3(1). = = "1 8" _4 3_ = A IA = "1 0" 1 "1 8" 4 3 = "1(1) + 0(4) -0(1) + 1(4) 1(8) + 0(3)" 0(8) + 1(3). = = "1 8" _4 3_ = A .". From(7) and (8 ) i VI = IA = A '1+0 + 8" 4 + + 3_ ...(7) 1+0 8 + 0" + 4 + 3_ ...(8) Example 1.4: If A = '2 3' 4 5. find A - 7A - 21 Solution: A = AA = A 2 = '2 3' 4 5. '2 3' 4 5. '4 + 12 6 + 15' .8 + 20 12 + 25. -7A = 21 = "16 21" _28 37_ -7 "2 3' _4 5. = '-14 _-28 -21" -35_ -2 "1 0" _0 1. = "-2 _ 0" -2_ (1) (2) (3) (1) + (2) + (3) gives A 2 - 7 A - 21 = A 2 + (- 7 A) + (- 21) 16 21" .28 37. 14 -21 28 -35. -2 i.e. ■ 7A - 21 = 16-14-2 21-21+0 .28-28 + 37-35-2. 0' .0 0. = o Example 1.5: If A = 1 4' .0 3. andB = '5 0' .3 9. show that (A + B) 2 * A 2 + 2AB + B 2 11 Solution: A + B = 1 4' .0 3. '5 0' .3 9. 1 + 5 4 + 0" .0 + 3 3 + 9. (A + B) z = (A + B) (A + B) = (A + B) 2 = ,2 6 4" .3 12. 6 4' .3 12. 6 4' 3 12. 36 +12 24 + 48' 18 + 36 12 + 144. '48 72" .54 156. (1) A" = A.A = B 2 = B.B = AB = 1 4' .0 3. 1 4' 3. 5 0" 3 9_ "5 .3 9 1 4 .0 3. 5 0" 3 9. 2AB =2 A 2 + 2AB + B 2 = A 2 + 2AB + B 2 = 5+12 + 36 + 9 + 27 34 72' 18 54. 25 0" 42 81. 1+0 4+12 0+0 0+9 "25 + + .15 + 27 + 81 17 36 "1 16" _0 9_ 0" "25 0" Sl_ = _42 81. 17 36 9 27. 1 161 [34 72 9_I + Ll8 54 60 88 60 144 From (1) and (2) we have (A + B) 2 * A 2 + 2AB + B 2 Example 1.6: Find the value of x if [2x 3] 9 27. 1 + 34 + 25 16 + 72 + 0" .0+18 + 42 9 + 54 + 81. (2) 1 2' 3 0. = Solution: [2jc-9 4jc + 0] = O (Multiplying on first two matrices) => [(2x - 9)x + 4x(3)] = O =^> [2jc 2 + 3x] = O i.e. 2jc 2 + 3x = => x(2x + 3) = 3 [2x A -9x+ \2x\ = O Hence we have x = 0, x = ~j~ Example 1. 7: Solve: X + 2Y = 4 6' 8 10. X-Y = 1 0' ■2-2. 12 Solution: Given X + 2Y = " 4 6" _-8 10. " 1 0" _-2 -2_ (1) - (2) X-Y = (X + 2Y) - (X - Y) = 3Y = Y = ...(1) ...(2) " 4 6" " 1 0" _-8 10. -2 -2 " 3 6" _-6 12. =* ■Y-i 3 6" -6 12. 1 2' 2 4. Substituting matrix Y in equation (1) we have X + 2 X + 1 21 [4 6" ■2 4j = L-8 10. 2 41 [ 4 6" ■4 8j = L-8 10. 4 61 2 41 '2 2' ■8 lOJ ~L-4 8j = L-4 2. 2 2" •4 2_ EXERCISE 1.1 (1) Construct a 3 x 3 matrix whose elements are (i) o« =i+j (ii) fly = i x 7 X = .-. x = andY = 1 2' •2 4. (2) Find the values of x, y, z if (3) If (4) If A = , , . B = -7 .3 2a. x 3x — y 2x + z 3y -w_ 2x 3x - y 2x + z 3y - w_ 2 1" 4 -2. following (i) - 2A + (B + C) (ii) A - (3B - C) (iii) A + (B + C) (iv) (A + B) + C (v)A + B (vi)B + A (vii)AB (viii) BA 3 2" 4 7_ ind x, y, z, w "4 - .1 -2 4_ and C = '-2 _ 1 -3 2 find each of the 13 " 1 2 3" "2 1 "1 1 -1" (5) Given A = -13 4 B = 2 -1 -2 and C = 2 1 -2 2 -1. .1 1-1. _1 -1 1. verify the following results: (i) AB * BA (ii) (AB) C = A(BC) (iii) A(B + C) = AB + AC '-2 13" "47 0" (6) Solve : 2X + Y H - 5-7 3 4 5 4_ = ; X-Y = -1 2 -6 _-2 8 -5. (7) If A= 3-5" -4 2_ 9 , show that A - 5 A - 14 I = O where I is the unit l natri> of order 2. (8) If A = "3 -2" _4 -2. "1 2 2" find k so that A 2 = kA- 21 (9) If A = 2 1 2 _2 2 1_ , show that A 2 - 4A - 51 = (10) Solve for x if 2 r c 1 2 3_ + 2x 3" 1 4_ = "3 4" _3 7_ "112" X (11) Solve for x if [x 2 -1] -1 -4 1 _- 1 - 1 - 2_ 2 .1. = [0] (12) If A = "1 2" 2 0_ B "3 ~_1 -1" 0. verify th e folio win g: (i) (A + B) 2 = A 2 + AB + BA + B 2 (ii) (A - B) 2 * A 2 - 2AB + B 2 (iii) (A + B) 2 * A 2 + 2AB + B 2 (iv) (A - B) 2 = A 2 - AB - BA + B 2 (v) A 2 - B 2 * (A + B) (A - B) (13) Find matrix C if A = '3 7' 2 5. B = ■3 2 4-1. and5C + 2B =A (14) IfA = 1 -1 2 -1. andB = and (A + B) 2 = A 2 + B 2 find x and y. 14 1.2 Determinants 1.2.1 Introduction: The term determinant was first introduced by Gauss in 1801 while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic forms. We know that the area of a triangle with vertices (jci, yi) (X2, yi) an d C*3, yj) is 2 [xi(y2-y3) + x2(y3-yi) + x3(yi-y2)] ■•■(!) Similarly the condition for a second degree equation in x and y to represent 2 2 2 a pair of straight lines is abc + 2fgh - af - bg - ch = . . . (2) To minimize the difficulty in remembering these type of expressions, Mathematicians developed the idea of representing the expression in determinant form. The above expression (1) can be represented in the form Similarly the second expression (2) can be expressed as Again if we eliminate x, y, z from the three equations a\x + b[y + c\ z = ; ci2X +b2y + C2Z = ; aye + b^y +cj,z = 0, we obtain ai(£>2 c 3 ~ ^3 c 2) ~ b\ (a.2 c 3 ~ a 3 c 2) + c l ( fl 2 ^3 _ a 3 i>2) = a\ b\ c\ x\ y\ 1 form 2 X2 yi i X3 y3 i a h g h b f = 0. 8 f c = 0. Thus a determinant is a particular This can be written as a 2 ^2 c 2 «3 b3 c 3 type of expression written in a special concise form. Note that the quantities are arranged in the form of a square between two vertical lines. This arrangement is called a determinant. Difference between a matrix and a determinant (i) A matrix cannot be reduced to a number. That means a matrix is a structure alone and is not having any value. But a determinant can be reduced to a number, (ii) The number of rows may not be equal to the number of columns in a matrix. In a determinant the number of rows is always equal to the number of columns. 15 (iii) On interchanging the rows and columns, a different matrix is formed. In a determinant interchanging the rows and columns does not alter the value of the determinant. 1.2.2 Definitions: To every square matrix A of order n with entries as real or complex numbers, we can associate a number called determinant of matrix A and it is denoted by I A I or det (A) or A. Thus determinant formed by the elements of A is said to be the determinant of matrix A. If A = «11 «12 «21 «22. then its I A I = an «12 «21 «22 = an a 22 _ #21«12 To evaluate the determinant of order 3 or above we define minors and cofactors. Minors: Let I A I = | [ajj] | be a determinant of order n. The minor of an arbitrary element a,y is the determinant obtained by deleting the i row and_/ column in which the element a„- stands. The minor of a„- is denoted by My. Cofactors: The cofactor is a signed minor. The cofactor of a,y is denoted by A y - and is defined as Ay = (- 1) ; +j My. The minors and cofactors of an, ai2, ^13 of a third order determinant are as follows: an a\2 an ail an an 031 «32 «33 an a 2 3 a-il «33 (i) Minor of aj 1 is Mn = Cofactor of an is An = (-1) (ii)Minor of «i2 is M[2 = : a 22«33 - a 32 «23- 1 + 1 Mn = an a 2 3 «32 «33 = «22«33 ~ a 32 <223 ail a 2 3 «31 «33 a 2i «33 -«31«23 1+2, Cofactor of a 12 is A 12 = (-l) M 12 = «21 «23 «31 «33 = - (a21«33 - «23 «3l) 16 (iii) Minor of a 13 is Mi 3 = a 2 \ an «31 a 32 : «21 «32~«31 a 22 1 + 3 Cofactorof an is A13 = (- 1) M13 = : «21 «32 - «31 «22 «21 «22 «31 (232 Note: A determinant can be expanded using any row or column as given below: a n «12 013 a 2 \ <222 023 031 «32 «33 LetA = A = an An + ai2 A12 + an A13 or a\\ Mn - ai2 M12 + ai3 M13 (expanding by Ri) A = a n An + «2lA2l + ^31 A 31 or «11 M ll ~ «21 M 2 i + 031 M 31 (expanding by Ci) A = a 2 \ A21 + (222 A22 + «23 A 23 or - (2 2 l M 2 i + «22 M 2 2 - «23 M 23 (expanding by R2) Example 1.8: Find the minor and cofactor of each element of the determinant 3 4 1 0-12 5-2 6 Solution: Minor of 3 is Mn = -1 -2 2 6 =-6+4=-2 Minor of 4 isMi2 = 2 5 6 = 0-10 = - 10 Minor of 1 is M13 = -1 5 -2 =0+5=5 Minor of is M21 = 4 1 -2 6 = 24 + 2 = 26 Minor of - 1 is M22 = 3 1 5 6 = 18-5 = 13 Minor of 2 is M23 = 3 A 5 - 2 = - 6 - 20 = - 26 17 Minor of 5 is M31 = Minor of - 2 is M32 = Minor of 6 is M33 = Cof actor of 3 is Aj 1 = Cof actor of 4 is A 12 = Cofactor of 1 is A13 = Cof actor of is A21 = Cofactor of - 1 is A22 = Cofactor of 2 is A23 = Cofactor of 5 is A3 1 = Cofactor of - 2 is A32 = Cofactor of 6 is A33 = Singular and non-singular matrices: 4 1 -1 2 3 1 2 3 =8+1=9 =6-0=6 4 -1 l + l 1+2 1 + 3 2+1 2 + 2 2 + 3 3 + 1 3 + 2 3 + 3 =-3-0=-3 Mn = Mn = -2 Mi2 = -M 12 =10 M 13 = Mi3 = 5 M 2 i = - M 2 i = - 26 M22 = M22= 13 M23 = - M23 = 26 M 3 i=M 31 =9 M32 = - M32 = - 6 M 33 = M33 = - 3 A square matrix A is said to be singular if I A I = A square matrix A is said to be non-singular matrix, if I A I ^ 0. is a singular matrix. 1 2 3" For example, A = 4 5 6 7 8 9_ 1 2 3 c v IAI = 4 5 6 = 1 8 7 8 9 -2 4 6 7 9 + 3 4 5 7 8 = 1(45 - 48) - 2 (36 - 42) + 3(32 - 35) = -3 + 12-9 = "1 7 5~ B = 2 6 3 L 4 8 9_ is a non-singular matrix. v 1 Bl = 1 2 4 7 5 6 3 8 9 = 1 6 3 8 9 -7 2 3 4 9 + 5 2 6 4 8 = 1(54 - 24) - 7(18 - 12) + 5 (16 - 24) = l(30)-7(6) + 5(-8) = - 52 * .". The matrix B is a non-singular matrix. 1.2.3 Properties of Determinants There are many properties of determinants, which are very much useful in solving problems. The following properties are true for determinants of any order. But here we are going to prove the properties only for the determinant of order 3. Property 1: The value of a determinant is unaltered by interchanging its rows and columns. Proof: LetA = Expanding A by the first row we get, A = ai(b2C 3 -b 3 C2)-bi(a2C 3 -a 3 C2) + c l (a2b 3 -a 3 b2) = a\b2Cj, - a\bj,C2 - a.2b\c 3 + a 3^1 c 2 + a 2^3 c l ~~ a 3^2 c l Cl\ b\ c\ «2 b 2 C2 «3 b 3 C3 Let us interchange the rows and columns of A. determinant. ...(1) Thus we get a new A, = a\ a.2 «3 b\ b 2 b 3 C[ c 2 C3 Since determinant can be expanded by any row or any column we get Ai = ai(&2C3 - C2&3) - ^1 («2C3 - C2a3) + ci(a2&3 - M3) = ai£>2 c 3 _ a \b^C2 - 02^1 c 3 + «3^lC2 + 02^3 C 1 _ 03^2 C 1 ■ • ■ (2) From equations (1) and (2) wehaveA = A[ Hence the result. Property 2: If any two rows (columns) of a determinant are interchanged the determinant changes its sign but its numerical value is unaltered. Proof: ci\ b\ c\ LetA = «2 «3 b 2 b 3 19 A = a\(b2 Co, - bj C2) - bi(ci2 Co, - C13 C2) + c\ (a2bi - 03 b2) A = ai^2 c 3 _a 1^3 c 2 _a 2^1 c 3 + fl 3^1 c 2 + fl 2^3 c l _fl 3^2 c l •••(!) Let Ai be the determinant obtained from A by interchanging the first and second rows. i.e. R[ and R2. ci2 b 2 c 2 Ai = a\ b\ c\ a-i b 3 c 3 Now we have to show that Ai = - A. Expanding Aj by R2, we have, Ai = -ai(b2C3-b 3 C2)+b l (a2C3-a 3 C2)-ci(a2b 3 -asb2) = -[aib2C3-aibT,C2 + a 2 bic 3 + a 3 b l C2 + a2b i c l -a3b2Ci] ...(2) From (1) and (2) we get Ai = - A. Similarly we can prove the result by interchanging any two columns. Corollary: The sign of a determinant changes or does not change according as there is an odd or even number of interchanges among its rows (columns). Property 3: If two rows (columns) of a determinant are identical then the value of the determinant is zero. Proof: Let A be the value of the determinant. Assume that the first two rows are identical. By interchanging Rj and R2 we obtain - A (by property2). Since Rj and R2 are identical even after the interchange we get the same A. i.e. A = -A => 2A = i.e. A = Property 4: If every element in a row (or column) of a determinant is multiplied by a constant "fe" then the value of the determinant is multiplied by k. Proof: LetA = Cl\ b\ c\ a 2 bi C2 «3 b 3 C3 20 Let A j be the determinant obtained by multiplying the elements of the first ka\ kb\ kci row by 'fe' then Aj = a 2 b2 c 2 a 3 b 3 c 3 Expanding along Ri we get, Aj = ka\(b2C 3 -b 3 C2)-kb\(ci2C 3 - a 3 C2) + kc\(fl2b 3 - a 3 b2) = fc[ai&2 c 3 ~~ a \b 3 C2 ~ a lb\c 3 + a 3 b\C2 + <22^3 C 1 ~~ a 3^2 c l] Ai = kA. Hence the result. Note: (1) Let A be any square matrix of order n. Then kA is also a square matrix which is obtained by multiplying every entry of the matrix A with the scalar k. But the determinant k IAI means every entry in a row (or a column) is multiplied by the scalar k. (2) Let A be any square matrix of order n then I kA I = fe n l A I. Deduction from properties (3) and (4) If two rows (columns) of a determinant are proportional i.e. one row (column) is a scalar multiple of other row (column) then its value is zero. Property 5: If every element in any row (column) can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows (columns) of both being the same. ai+*i Pi+ji yi + zi Proof: Let A = by b 2 b 3 C{ c 2 c 3 Expanding A along the first row, we get A = (ai + x\) b2 b-i ci c 3 (Pi+yi) by 63 c\ c 3 + (Yl + zi) c\ in = <M b 2 b 3 C2 c 3 P b 3 C3 + Y1 + Ui in C2 b 2 b 3 C2 c 3 b\ b 3 b\ b 2 -y\ + z\ c\ c 3 c\ c 2 21 xi y\ Z\ by b 2 h c\ c 2 C3 ai pi yi b\ b 2 b 3 c\ c 2 C3 Hence the result. Note: If we wish to add (or merge) two determinants of the same order we add corresponding entries of a particular row (column) provided the other entries in rows (columns) are the same. Property 6: A determinant is unaltered when to each element of any row (column) is added to those of several other rows (columns) multiplied respectively by constant factors. i.e. A determinant is unaltered when to each element of any row (column) is added by the equimultiples of any parallel row (column). Proof: ci\ b\ c\ LetA= a 2 b 2 c 2 a 3 bi C3 Let Aj be a determinant obtained when to the elements of Ci of A are added to those of second column and third column multiplied respectively by I and m. a\ + lb\ + mc\ A]= a 2 + lb 2 + mc 2 aj, + Ibj, + uct, a\ b\ c\ a 2 b 2 c 2 a-i h c 3 a\ b\ c\ a 2 b 2 c 2 aj, b 3 c 3 Therefore Aj = A. 01 b 2 b 3 \b { c\ C2 C3 'l c\ lb 2 b 2 c 2 lbs h ct, mc\ b\ c\ mc 2 b 2 C2 mc-i, b 3 C3 (by property 5) + + Ci is proportional to C 2 in the second det. Ci is proportional to C3 in the third det. Hence the result. Note: (1) Multiplying or dividing all entries of any one row (column) by the same scalar is equivalent to multiplying or dividing the determinant by the same scalar. 22 (2) If all the entries above or below the principal diagonal are zero (upper triangular, lower triangular) then the value of the determinant is equal to the product of the entries of the principal diagonal. For example, let us consider 3 2 7 I A I = 5 3 =3(5-0) -2(0-0) + 7(0-0) = 15 1 The value of the determinant A is 15. The product of the entries of the principal diagonal is 3 x 5 x 1 = 15. x - 1 x x -2 x-2 x-3 =0 x-3 Solution: Since all the entries below the principal diagonal are zero, the value of the determinant is (x - 1) (x - 2) (x - 3) .-. (x- l)(x-2)(x-3) = =^ x=l, x = 2, x = 3 x 5 1 -1 = 36 = Example 1.9: Solve Example 1.10: Solve for x if x 5 7 x + 1 -2 -1 1 Solution : x 5 7 x + 1 -2 -1 1 = => (x 2 -35) + (l-2) = => jc 2 - 35 - 1 => x 2 = 36 => x = i 6 Example 1.11: Solve for x if Solution: <°> 3 x - 1 1 x + <°) -x +x = i.e. x(l — x) = Example 1.12: Evaluate (i) Solution: (i) Let A = 1 A' 2 X 1 3 X = = x = 0, x = 1 0-l[x -x] + = a £> + c b c + a (ii) c a + b x + 2a x + 3a x + Aa x + 3a x + Aa x + 5a x + Aa x + 5a x + 6a a b + c b c + a = c a + b 1 a a + b + c 1 b a + b + c 1 c a+b + c c 3 ^ c 3 + c 2 23 = [v Ci is proportional to C3] (ii) Let A = = Example 1.13: Solution: x + 2a x + 3a x + Aa x + 2a a x + 3a x + 4a x + 5a = x+ 3a a x + Aa x + 5a x + 6a x + Aa a 2a 2a 2a C 2 -> C 2 c 3 -> c 3 Ci ■Ci [ .• C 2 is proportional toC 3 ] 2x + y x y Prove that 2y + z y z = 2z + x Z X 2x + y x y 2x x y y x y 2y + z y z — 2y y z + z y z 2z + x Z X 2z z x X z X Ciis proportional to C 2 in the first det." J + u '•'Ci is identical to 1 "3 in the second det. _ = Example 1.14: Prove that Solution: 1 a a 2 1 b b 2 1 c 2 c = (a -b) (b - c) (c - a) 1 a a 2 1 b b 2 1 c c 2 a-b a L -b L Ri -> Ri - R2 — b-c b 2 -c 2 1 c c 2 R 2 -> R2 - R3 1 a + b Take (a - b) and (b - -c) = (a-b)(b- c) 1 b + c from Ri and R 2 1 c c 2 respectively. Example 1.15: Solution: = (a-b) (b-c) [(1) (b + c)- (1) (a + b)] = (a-b) (b-c) (c-a) ■■xy 1 1 1 "ove that 1 l+x I 1 1 l+y 1 1 1 1 1 l+x I = 1 I I + y y R 2 -> R2 R3 -> R3 Ri Ri = xy [ ".■ upper diagonal matrix] 24 Example 1.16: Prove that Mb 2 Me 2 be b + c ca c + a ah a + b = \lar Mb 2 Mc 2 be b + c ca c + a ab a + b 1 abc abc abc 1 abc 1 abc (ab+bc+ca) abc Ma abc a(b + c) Mb abc b(c + a) Mc abc c(a + b) Ma 1 a(b + c) Mb 1 b(c + a) Mc 1 c(a + b) be 1 a(b + c) ca 1 b(c + a) ab 1 c(a + b) be 1 ab + be + ca ca 1 ab + be + ca ab 1 ab + be + ca be 1 1 ca I I ab 1 1 Multiply R 1; R 2 , R3 by a, b, c respectively Take abc from C2 Multiply Ci by abc C 3 -> C 3 + Ci Example 1.17: Prove that Solution: Let A = (ab + be + ca) ,„ N abc = W ,2 2 , C DC b + c 5 that 2 2 c a ca c + a a b ab a + b ^ 2 be b + c c 2 « 2 ca c + a « 2 ^ 2 ab a + b Take (ab + bc + ca) from C3 [ v C2 is identical to C3] = Multiply Ri, R2 and R3 by a, o and c respectively A = a&c 2 2 ai> c abc ab + ac 2 2 be a abc be + ab ca b abc ca + be 25 (abc) abc be 1 ab + ac ca 1 be + ab ab 1 ca + be Take abc from Ci and C 2 be 1 ab + be + ca = abc ca 1 ab + be + ca ab 1 ab + be + ca be 1 1 = abc (ab + be + ca) ca 1 1 ab 1 1 = abc (ab + be + ca) (0) = a+b+c -c C 3 -> C 3 + Ci Take (ab + be + ca) from C3 [ v C 2 is identical to C3] Example 1.18 : Prove that Solution: a+b+c -c -b -c a+b+c -a —b —a a+b+c -c a+b+c -a -b -a a+b+c = 2(a+b) (b+c) (c+a) a + b a + b - (a + b) - (b + c) b + c b + c — b -a a + b + c Ri ->Ri + R 2 R 2 -> R2 + R 3 -1 = (a + b) (b + c) = (a + b)(b + c) = (a + b)(b + c)x (- 2) 1 1 -1 1 1 -b -a a+b+c 2 -11 1 - b -a a + b + c 1 1 b a+b+c = (a + b)(b + c)x (- 2) [- (a + b + c) + b] = (a + b)(b + c)x (- 2) [- a - c] A = 2(a + b) (b + c) (c + a) Take (a + b), (b + c) from Ri and R 2 respectively Ri -> R t + R 2 Example 1.19: Prove that 2 , i a + k ab ac ab b 2 + k be ac be c 2 + k = k 2 (a 2 + b 2 + c 2 + k) 26 Solution: Let A = a 2 + X ab ab b 2 + X cu- be c + X ac be Multiply Ri, R2 and R3,by a, b and c respectively A = 1 abc a(a + X) a b ab 2 b(b 2 + X) 2 a c ac be c(c + X) Take a, b and c from Ci, C2 and C3 respectively A = abc abc a 2 + X b 2 ,2 b 2 + X c 2 + X a 2 +b 2 +c 2 +X a 2 +b 2 +c 2 +X a 2 +b 2 +c 2 +X = (a 2 + b 2 + c 2 + X) = (a 2 + b 2 + c 2 + X) = (a 2 + b 2 + c 2 + X) b 2 +X c 2 +X Ri-> Ri + R 2 + R3 1 1 1 b 2 b 2 + X b 2 c 2 + X c c 1 b 2 X c 2 X X X C 2 ^C 2 -C! C 3 ^C 3 -C! 2 , 1 a + X ab ac ab ac b 2 + X be be c 2 + X = X\a 2 + b 2 + c 2 + X) EXERCISE 1.2 2 (1) Find the value of the determinant expansion. 15 10 without usual 27 (2) Identify the singular and non-singular matrix "1 4 9" ' 1 2 3 " (i) 4 9 16 _9 16 25_ (ii) 4 5 6 _- 2 - 4 -6. 2 jc 4 4 3 9 (3) Solve (i) 3 2 1 1 2 3 = ~ 3 (ii) 3-2 7 4 4 jc = -1 a - b b - c c - a 1 ab c(a + b) (4) Evaluate (i) b - c c - a a- b (ii) 1 be a(b + c) c - a a - b b - c 1 ca b(c + a) a - b - c 2a 2a (5) Prove that 2b b - c - a 2b 2c 2c c - a - b = (a + b + c) 3 l + a 1 1 = abc(\+l + l + ^\ (6) Prove that 1 \+b 1 1 1 1 + c where a, b, c are non zero real numbers and hence evaluate the l + a 1 1 value of 1 l+a 1 1 1 l + a 1 a a (7) Prove that 1 b b 3 1 c c 3 = (a - b) (b - c) (c - a) (a + b + c) XX 1 — X (8) If x, y, z are all different and 2 1 3 y y \-y 2 1 3 z z \-z = then show that xyz = 1 1 2 1 a a I a be (9) Prove that (i) 1 b b 2 -- 1 2 lcc - \ b ca 1 c ab y + Z x y (ii) Z + X Z X = (x + y + z) (x-zj 2 x + y y z 28 (10) Prove that b+c c+a a+b q+r r+p p+q y+z z+x x+y ' b c • c a =3 a a b a b c a - b b - c c - a • + c c + a a + b (i) (iii) (iv) a b c =2 p q r x y z (ii) -a ab ac ab -b be 2 ac be -c = 4a 2 b 2 c 2 = 3abc -a —b — c = a + b + c — 3abc 1.2.4 Factor method Application of Remainder theorem to determinants Theorem: If each element of a determinant (A) is polynomial in x and if A vanishes for x = a then (x - a) is a factor of A. Proof: Since the elements of A are polynomial in x, on expansion A will be a polynomial function in x. (say p(x)). For x = a, A = i.e. p(x) = when x = a, i.e. p(a) = .". By Remainder theorem (x - a) is a factor of p(x). i.e. (x - a) is a factor of A. Note: (1) This theorem is very much useful when we have to obtain the value of the determinant in 'factors' form. Thus, for example if on putting a = b in the determinant A any two of its rows or columns become identical then A = and hence by the above theorem a - b will be a factor of A. (2) If r rows (column) are identical in a determinant of order n (n > r) when we put x = a, then (x - a) r ~ is a factor of A. (3) (x + a) is a factor of the polynomial fix) if and only if x = - a is a root of the equation/(x) = 0. 29 Remark: In this section we deal certain problems with symmetric and cyclic properties. Example 1.20: Prove that c c = (a- b) (b - c) (c - a) (a + b + c) Solution: 1 a a LetA = 1 b b 3 1 c c 3 .Puta = b, A = 1 1 lcc = [ v Ri is identical to R2] Thus a a b b 3 „3 = (a - b) (b - c) (c - a) k(a + b + c) .'. (a - b) is a factor of A. Similarly we observe that A is symmetric in a, b, c, by putting b = c, c = a, we get A = 0. Hence (b - c) and (c - a) are also factors of A. .". The product (a-b) (b-c) (c - a) is a factor of A. The degree of this product is 3. The product 3 of leading diagonal elements is 1 . b . c . The degree of this product is 4. .". By cyclic and symmetric properties, the remaining symmetric factor of first degree must be k(a + b + c), where k is any non-zero constant. 1 1 1 c c~ To find the value of k, give suitable values for a, b, c so that both sides do not become zero. Take a = 0, b= 1, c = 2. = fe(3)(-l)(-l)(2) => k=l :. A = (a- b) (b - c) (c -a) (a + b + c) Note: An important note regarding the remaining symmetric factor in the factorisation of cyclic and symmetric expression in a, b and c If m is the difference between the degree of the product of the factors (found by the substitution) and the degree of the product of the leading diagonal elements and if (1) m is zero then the other symmetric factor is a constant (k) (2) m is one then the other symmetric factor of degree 1 is k(a + b + c) (3) m is two then the other symmetric factor of degree 2 is 2 2 2 kia + b + c )+l (ab+bc+ca) 1 1 1 1 1 2 8 30 Example 1.21: Prove by factor method Solution: 1 2 3 \ a a 1 2 3 = (a- b) (b -c) (c-a) (ab + be + ca) LetA = 2 3 I a a 1 a ui 1 c 2 c 3 Put a=b A = 1 c 2 c 3 = [vRi = R 2 ] .". (a - b) is a factor of A. By symmetry on putting b = c and c = a we can easily show that A becomes zero and therefore (b - c) and (c - a) are also factors of A. This means the product (a - b) (b - c) (c - a) is a factor of A. The degree 2 3 of this product is 3. The degree of the product of leading diagonal elements b c is 5. 2 2 2 .". The other factor is k(a + b + c ) + l(ab + be + ca) 1 2 3 1 a a 1 b 2 b 3 1 c 2 c 3 = [k(a 2 + b 2 + c 2 ) + l(ab + be + ca)] (a - b) (b - c) (c - a) To determine k and I give suitable values for a, b and c so that both sides do not become zero. Take a = 0, b = 1 and c = 2 = [k (5) + 1(2)] (- 1) (- 1) (2) 1 1 1 1 1 4 8 4 = (5k + 21) 2 5k + 21 = 2 ...(1) Again put a = 0, b = - 1 and c = 1 1 11-1 = [k(2) + /(- 1)] (+1)(-2)(1) 1 1 1 => 2 = (2k - I) (- 2) ^ 2fc - Z = - 1 On solving (1) and (2) we get k = and Z = 1 ...(2) 31 .". A = (ab + be + ca) (a - b) (b - c) (c - a) = (a - b) (b - c) (c - a) (ab + be + ca) Example 1.22: Prove that Solution: (b + cf u2 a (c + a) 2 2 (a + by = 2abc (a + b + c)~ Let A = (b + cf ,2 A = (b + c) ,2 (c + a) 2 Put a = we get 2 j2 c b a (a + by = [ v C2 is porportional to C3] c c .'. (a - 0) = a is a factor of A. Similarly on putting b = 0, c = 0, we see that the value of A is zero. .". a, b, c are factors of A. Put a + b + c = 0, we have A = (-aY b 2 2 (-by (-cY = 2 ■ . Since three columns are identical, (a + b + c) is a factor of A. 2 .'. abc (a + b + c) is a factor of A and is of degree 5. The product of the leading diagonal elements (b + c) (c + a) (a + b) is of degree 6. .". The other factor of A must be k(a + b + c). (b + c y u2 (c + a) 2 = k abc (a + b + c)~ c c (a + b) Take the values a = 1, b = 1 and c = 1 4 1 1 1 4 1 1 1 4 = k(l) (I) (I) (3)" 54 = 27k k = 2 .-. A = 2abc (a + b + c)~ 32 Example 1.23: Show that Solution: Let A = X a a a X a a a X X a a a X a a a X Put x = a = (x - a) ix + 2d) :. A = \2- a a a a a a a a a = Since all the three rows are identical (x -a) is a factor of A. Put x = - 2a. ■2a a a A = a -2a a a a —2a (x + 2d) is a factor of A. a a -2a a a -2a = [C 1 ^C 1 + C 2 + C 3 ] .". (x -a) (x + 2a) is a factor of A and is of degree 3. The degree of the product of leading diagonal element is also 3. Therefore the other factor must be k. = k(x -a) (x + 2a). X a a a X a a a X Equate x term on both sides, 1 = k Example 1.24: Using factor method, prove Solution: Put;c=l, A = Let A = 2 3 5 2 3 5 2 3 5 jc+1 3 2 x+2 2 3 = A' a a a X a = (- a a X x+l 3 5 2 x+2 5 2 3 x+4 5 5 A' + 4 = (x - a) ix + 2d) = (x-iy ( x + 9) Since all the three rows are identical, (x - 1) is a factor of A. 33 Put x = -9 in A, then A = 3 5 = 0-7 5 3 -5 = [vC^d+Ci+Cg] -8 3 5 2-7 5 2 3-5 .'. (x + 9) is a factor of A. The product (x - 1) (x + 9) is a factor of A and is of degree 3. The degree of the product of leading diagonal elements (x + 1) (x + 2) (x + 4) is also 3. .". The remaining factor must be a constant "fe" x+l 3 5 2 x + 2 5 = fc(x - l) 2 (x + 9). Equating x 3 term on both 2 3 jc + 4 sides we get k = 1 Thus A = (x - l) z (x + 9) (1) Using factor method show that (2) Prove by factor method (3) Solve using factor method EXERCISE 1.3 I a a 1 lcc b + c a - c a - b b - c c + a b - a c - b c - a a + b x + a b c = (a - b) (b - c) (c - a) = 8abc a a x + b c b x + c = (4) Factorise ■ 2 2 i c (5) Show that = (a + b + c) (a- b) (b - c) (c - a) be ca ab b + c a a" c + a a + b c c z 1.2.5 Product of determinants Rule for multiplication of two determinants is the same as the rule for multiplication of two matrices. While multiplying two matrices "row-by-column" rule alone can be followed. The process of interchanging the rows and columns will not affect the value of the determinant. Therefore we can also adopt the following procedures for multiplication of two determinants. 34 (1) Row-by-row multiplication rule (2) Column-by-column multiplication rule (3) Column-by-row multiplication rule Note: The determinant of the product matrix is equal to the product of the individual determinant values of the square matrices of same order. i.e. Let A and B be two square matrices of the same order. We have I AB I = I A I I B I This statement is verified in the following example. Example 1.25: If A = cos9 - sin6 _sin6 cos8 . ,B = cos8 sin8 - sin9 cos0. are two square matrices then show that I AB I = I A I I B I Solution: Given that A = AB = cos9 - sin9 sin9 cos9 cos9 - sin9 sin9 cos9 cos 2 9 + sin 2 G andB = cos9 sin9 - sin9 cos9. cos9 sin9 - sin9 cos9. cos9 sin0 - sin0 cos9 IABI = IAI = IBI = _sin0 cos0 - cos0 sin0 1 cos + sin 6 1 cos9 sin9 cos9 - sin9 = 1 - sin9 cos9 sin9 cos9 = cos 2 9 + sin 2 9 = 1 = cos 2 9 + sin 2 9 = 1 IAI I B 1= 1x1 = 1 From (1) and (2) Example 1.26: Show that Solution: L.H.S. = I AB I = I A I IBI c b 2 c o a = b a ,2 , 2 7 + C ab ab 2 2 c + a be ac be a +1 c b 2 c b c b c a = c a c a b a o b a b a o + c + b o + o + ab 2 2 o + o + ab c + o + a o + ac + o be + o + o o + ac + o be + o + o b 2 + a 2 + (1) (2) 35 2 , i2 c + b ab ac ab 2 2 c +a ac be *2 , 2 b + a = R.H.S. Example 1.27: Prove that Solution: L.H.S. = a\ b\ a 2 b2 a\ b\ a 2 b2 a\ a 2 by b 2 a\ b\ 02 b 2 a\ b\ a 2 b 2 be 2 2 a\ + <22 a\b\ + a 2 b 2 2 2 aj&l + (22^2 ^1 + i>2 a\ b\ a 2 b 2 /^Interchange rows and Vcolumns of the first determinant 2 2 a\ + a 2 a\b\ + a 2 b 2 a\b\ + a 2 b 2 by + b 2 Example 1.28: Show that Solution: R.H.S. = 2bc — a c 2 c 2ca - b 2 a b 2 a 2 lab — c a b c = b c a c a b a b c 2 a b c a b c b c a = b c a b c a c a b c a b c a b a b c a b c = b c a x(- -1) c a b ' c a b t ? c a Interchanging R2 and R3 in the 2 nd determinant a b c b c a cab - a +bc + cb 2 — ab + c + ab , *2 - ac + ac + b 2bc - a ,2 - b - c a b c a 2 - ab + ab + c 2 - b + ac + ac -be + a + be , j2 ac + b + ac be + be 2 + a c + ab + ba c z b z 2ac - b a 9 9 a lab - c = L.H.S. 36 1.2.6 Relation between a determinant and its co-factor determinant a\ b\ c\ Consider A = a 2 b 2 c 2 a?, bj, c 3 Let Ai, Bi, Ci be the co-factors of a\, b\,c\ in A .". The cofactor determinant is Ai Bi C -l t is A 2 B 2 C 2 A3 B 3 C 3 .-. A = = Cl\ b 2 c 2 h c 3 ■b\ a 2 c 2 «3 c 3 + C\ a 2 b 2 aj, bj, Let A be expanded by Ri => A = a\ (co-factor of a\) + b\ (co-factor of b\) + c\ (co-factor of c\) => A = a\A\ + by Bj + c\ <Z\ i.e. The sum of the products of the elements of any row of a determinant with the corresponding row of co-factor determinant is equal to the value of the determinant. Similarly A = a 2 A 2 + b 2 B 2 + c 2 C 2 A = C13A3 + & 3 B 3 + c 3 C 3 Now let us consider the sum of the product of first row elements with the corresponding second row elements of co-factor determinant i.e. let us consider the expression aiA 2 + &iB 2 + ciC 2 = - a\ b\ c\ a\ c\ a\ b\ b 3 c-i + h a 3 C3 -c\ «3 b 3 a-sb\) = - ai(biCT, - b^cx) + b\(a\c?, - a 3 ci) - c\(a\b?, ■ = .". The expression a\A 2 + b\B 2 + c\C 2 = Thus we have aiA 3 + ^iB 3 + C1C3 = ; a 2 Ai + b 2 B[ + c 2 C\ = ; a 2 A 3 + & 2 B 3 + c 2 C 3 = a 3 Ai +£ 3 Bi +c 3 Ci =0 ; a 3 A 2 + £ 3 B 2 + c 3 C 2 = i.e. The sum of the products of the elements of any row of a determinant with any other row of co-factor determinant is equal to Note: Instead of rows, if we take columns we get the same results. .". A = a[A\ + a 2 A 2 + a 3 A 3 A = friBi + £> 2 B 2 + £ 3 B 3 A = ciCi + c 2 C 2 + c 3 C 3 Thus the above results can be put in a tabular column as shown below. 37 Row-wise Column-wise Ri R 2 R 3 Ci c 2 c 3 n A c\ A n A ci A n A C3 A a\ b\ c\ a 2 b 2 ci «3 h Ci Where r,'s q's are i row and i column of the original determinant R/s, Q's are i row and i column respectively of the corresponding co-factor determinant. Example 1.29: If Ai , B i, Ci are the co-factors of a\, b\, c\ in A = Ai Bi Ci then show that A 2 B 2 C 2 = A z A 3 B 3 C 3 Ai Bi Ci Solution: «2 ^2 c 2 A 2 B 2 C 2 A 3 B 3 C 3 a\A\ + b\B[ + c\C\ «iA 2 + i>iB 2 + cjC 2 aiA 3 + £>iB 3 + cjC 3 a 2 Ai + i> 2 Bj + c 2 Ci a 2 A 2 + £> 2 B 2 + cjC 2 a 2 A 3 + £> 2 B 3 + c 2 C 3 a 3 Ai + & 3 Bi + c 3 Ci a 3 A 2 + 6 3 B 2 + c 3 C 2 a 3 A 3 + 6 3 B 3 + c 3 C 3 A a\ b\ c\ a 2 b 2 c 2 a 3 bj, c 3 A A = A J i.e. A x A] Bi Ci A 2 B 2 C 2 A 3 B 3 C 3 = A J A, Bi Ci A 2 B 2 c 2 A 3 B 3 c 3 = A Z EXERCISE 1.4 1 a a 2 , ~ 2 2 2 1 - 2a - a - a (1) Show that a 1 a — - a - 1 a -2a a a 1 2 2- , - a a —2a — 1 1 X 2 A' a 1 2a (a — jc) (£> — x) (2) Show that 1 y y 2 b 2 1 2b = (a-y) 2 (b-y) 2 1 z 2 Z c 2 1 2c 2 (a-z) (b-* -:f (c-x) (c-y) 2 (c-z) 2 38 2. VECTOR ALGEBRA 2.1 Introduction: The development of the concept of vectors was influenced by the works of the German Mathematician H.G. Grassmann (1809 - 1877) and the Irish mathematician W.R. Hamilton (1805 - 1865). It is interesting to note that both were linguists, being specialists in Sanskrit literature. While Hamilton occupied high positions, Grassman was a secondary school teacher. The best features of Quaternion Calculus and Cartesian Geometry were united, largely through the efforts of the American Mathematician J.B. Gibbs (1839 - 1903) and Q. Heariside (1850 - 1925) of England and new subject called Vector Algebra was created. The term vectors was due to Hamilton and it was derived from the Latin word 'to carry'. The theory of vectors was also based on Grassman' s theory of extension. It was soon realised that vectors would be the ideal tools for the fruitful study of many ideas in geometry and physics. Vector algebra is widely used in the study of certain type of problems in Geometry, Mechanics, Engineering and other branches of Applied Mathematics. Physical quantities are divided into two categories - scalar quantities and vector quantities. Definitions: Scalar : A quantity having only magnitude is called a scalar. It is not related to any fixed direction in space. Examples : mass, volume, density, work, temperature, distance, area, real numbers etc. To represent a scalar quantity, we assign a real number to it, which gives its magnitude in terms of a certain basic unit of a quantity. Throughout this chapter, by scalars we shall mean real numbers. Normally, scalars are denoted by a, b,c... Vector : A quantity having both magnitude and direction is called a vector. Examples : displacement, velocity, acceleration, momentum, force, moment of a force, weight etc. Representation of vectors: Vectors are represented by directed line segments such that the length of the line segment is the magnitude of the vector and the direction of arrow marked at one end denotes the direction of the vector. 39 A vector denoted by a = AB is determined by two points A, B such that the magnitude of the vector is the length of the A Fig. 2. 1 line segment AB and its direction is that from A to B. The point A is called initial point of the vector AB and B is called the terminal point. Vectors are -> -> -» generally denoted by a , b , c ... (read as vector a, vector b, vector c, ...) Magnitude of a vector -> — > The modulus or magnitude of a vector a = AB is a positive number which is a measure of its length and is denoted by I a I =1 AB I = AB The modulus of a is also written as 'a' Thus ul = a ; Im =fc ; l~cl = I AB I = AB ; I CD I = CD ; I PQ I = PQ Caution: The two end points A and B are not interchangeable. Note: Every vector AB has three characteristics: Length : The length of AB will be denoted by I AB I or AB. Support : The line of unlimited length of which AB is a segment is called the support of the vector AB , Sense : The sense of AB is from A to B and that of BA is from B to A. Thus the sense of a directed line segment is from its initial point to the terminal point. Equality of vectors: -» -» -» — » Two vectors a and b are said to be equal, written as a = b , if they have the (i) same magnitude (ii) same direction 40 In fig (2.2) AB II CD and AB = CD AB and CD are in the same direction > — > -> -> Hence AB = CD or a = b 2.2 Types of Vectors Zero or Null Vector: A vector whose initial and terminal points are coincident is called a zero or -> null or a void vector. The zero vector is denoted by O Vectors other than the null vector are called proper vectors. Unit vector: A vector whose modulus is unity, is called a unit vector. The unit vector in the direction of a is denoted by a (read as 'a cap'). Thus \a\ = 1 The unit vectors parallel to a are + a ult: a direction)] -> -» A Result: a = \ a \ a [i.e. any vector = (its modulus) x (unit vector in that Itl ; ("a *o) In general vector in that direction unit vector in any direction = modulus of the vector Like and unlike vectors: Vectors are said to be like when they have the same sense of direction and unlike when they have opposite directions. a c like vectors unlike vectors Fig. 2. 3 41 Co-initial vectors: Vectors having the same initial point are called co-initial vectors. Co-terminus vectors: Vectors having the same terminal point are called co-terminus vectors. Collinear or Parallel vectors: Vectors are said to be collinear or parallel if they have the same line of action or have the lines of action parallel to one another. Coplanar vectors: Vectors are said to be coplanar if they are parallel to the same plane or they lie in the same plane. Negative vector: -» The vector which has the same magnitude as that of a but opposite -> -> — > -> direction is called the negative of a and is denoted by - a . Thus if AB = a > -> then BA = - a . Reciprocal of a vector: Let a be a non-zero vector. The vector which has the same direction as that of a but has magnitude reciprocal to that of a is called the reciprocal of a and is written as V, a ) where \\a ) 111 ~~ a Free and localised vector: When we are at liberty to choose the origin of the vector at any point, then it is said to be a free vector. But when it is restricted to a certain specified point, then the vector is said to be localised vector. 2.3 Operations on vectors: 2.3.1 Addition of vectors: Let OA = a , AB = b Join OB. Then OB represents the addition (sum) of the vectors a and b . This is written as OA + AB = OB Thus OB = OA + AB = a + b 42 This is known as the triangle law of addition of vectors which states that, if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order. Applying the triangle law of addition of vectors in A AABC, we have BC + CA = BA => BC + CA = - AB B C => AB+BC+CA="o' Fig. 2. 5 Thus the sum of the vectors representing the sides of a triangle taken in order is the null vector. Parallelogram law of addition of vectors: a -> If two vectors a and b are represented in magnitude and direction by the two adjacent sides c of a parallelogram, then their sum c is represented by the diagonal of the parallelogram which is co-initial with the given vectors. Symbolically we have "OP + OQ =OR r .. , , Fig. 2. 6 Thus if the vectors are represented by two adjacent sides of a parallelogram, the diagonal of the parallelogram will represent the sum of the vectors. By repeated use of the triangle law we can find the sum of any number of vectors. > -> Let OA = a . -» BC =~c . CD = d AB = be any five vectors as shown in the fig (2.7). We observe from the figure that each new vector is drawn from the terminal point of its previous one. OA + AB + BC + CD + DE = OE Thus the line joining the initial point of the first vector to the terminal point of the last vector is the sum of all the vectors. This is called the polygon law of addition of vectors. 43 Note : It should be noted that the magnitude of a + b is not equal to the sum -> -> of the magnitudes of a and b . 2.3.2 Subtraction of vectors: If a and & are two vectors, then the subtraction of & from a is -> -> -> -> defined as the vector sum of a and - b and is denoted by a - o . a — b = a ♦ (-*) Let OA = a and AB = ->• Then OB = OA + AB = a + b To subtract & from a , produce BA to B' such that AB = AB'. .-. AF? = - AB = - b a — b Now by the triangle law of addition OB' = OA + AB' = a + \- b ) = Properties of addition of vectors: Theorem 2.1: -> -> Vector addition is commutative i.e., if a and b are any two vectors then -» -» -» -» > -> > -> Let OA = a AB = b C t B InAOAB, OA + AB = OB rJ ' /I (by triangle law of add.) 7 s^ Jr =^> a + b = OB ... (1) P , / Complete the parallelogram OABC O 7J* A CB = OA ="? ; OC = AB =~? (J Fig. 2. 9 In AOCB, we have OC + CB = OB i.e. => b + "a =OB ...(2) -> -» -» -» From (1) and (2) we have a + £ = & + a .". Vector addition is commutative. 44 Theorem 2.2: Vector addition is associative -> -» -> i.e. For any three vectors a , b , c Proof : > -> Let OA = a Join O and B In AOAB, In AOBC, In AABC, In AOAC AB = OandC -> BC = c AandC OA + AB = -» -> a + b = OB + BC = /-> -A -> AB + BC = ft + c = OB OB OC oc AC AC OC OC (1) (2) [using (1)] (3) (4) [using (3)] OA + AC = -> (-> ->^ => a+^£+c,/ = From (2) and (4), we have \a + b ) + c = a + \b + c ) :. vector addition is associative. Theorem 2.3: -> -» -» — » — » -» -> For every vector a, a +0=0+a=a where O is the null vector, [existence of additive identity] Proof: Let OA = a Then Also a + O = OA + AA = OA = a .". a + O = a O + a =00 + OA = OA = a 45 .". O + a = a -> -» -» -» -> .". a+0=0+a=a Theorem 2.4: -> -> For every vector a , there corresponds a vector - a such that a +^- a ) = O =\— a )+a [existence of additive inverse] Proof: LetOA = a . ThenAO =- a -> / -A > > > -> .-. a + ^- a ^ = OA + AO = OO = O (""«) + a = AO + OA = AA = O Hence a + ^- a / =^-a/+a =0 2.3.3 Multiplication of a vector by a scalar Let m be a scalar and a be any vector, then ma is defined as a vector having the same support as that of a such that its magnitude is I m I times the magnitude of a and its direction is same as or opposite to the direction of a according as m is positive or negative. Result : Two vectors a and b are collinear or parallel if and only if a = mb for some non-zero scalar m. -» For any vector a we define the following: -> -» -> -> -> — > (1) a = a ; (- 1) a = - a ; Oa = O Note: If a is a vector then 5 a is a vector whose magnitude is 5 times the -> -> -» magnitude of a and whose direction is same as that of a . But -5a is a vector whose magnitude is 5 times the magnitude of a and whose direction is -> opposite to a . Properties of Multiplication of vectors by a scalar The following are properties of multiplication of vectors by scalars. -> -> For vectors a , b and scalars m, n we have 46 (i) wz\— a ) =(—m) a = - \m a ) (iii) ot^m a ) = (mn) a = n\m a ) (ii) (-m) \— a ) = m a -> -> -» (iv) (m + n) a =ma + n a Theorem 2.5 (Without Proof) If a and b are any two vectors and m is a scalar then wz\ a + b ) =ma + mb . -> /-> -A Result : m^ a - b ) = i 2.4 Position vector If a point O is fixed as the origin in space (or plane) and P is any point, then OP is called the position vector (P.V.) of P with respect to O. > -> From the diagram OP = r Similarly OA is called the position vector (P.V.) of A with respect to O and OB is the P.V. of B with respect to O. Fig. 2.11 Theorem 2.6: AB = OB respectively. OA where OA and OB are the P. Vs of A and B -> -» Proof: Let O be the origin. Let a and b be the position vectors of points A and B respectively Then OA = a ; OB = b In AOAB, we have by triangle law of addition OA + AB = OB => AB = OB i.e. AB = (P.V of B)- (P.V of A) — > -> -> OA = b - a Fig. 2. 12 47 Note : In AB , the point B is the head of the vector and A is the tail of the vector. .-. AB = (P. V. of the head) - (P. V. of the tail). Similarly BA = OA - OB The above rule will be very much useful in doing problems. Theorem 2.7: [Section Formula - Internal Division] -> -> Let A and B be two points with position vectors a and b respectively and let P be a point dividing AB internally in the ratio m : n. Then the position vector of P is given by OP = -> -» n a + mb m + n Proof: Let O be the origin. Then OA = a OB = -> -> > -> Let the position vector of P with respect to O be r i.e. OP = r Let P divide AB internally in the ratio m : n AP m Then PB _ n n AP = m PB m(oP-Oa) =m(oB-Op) n r - n a = mb — m r -» -» -> (m + n) r = mb + n a n AP = m PB => n \ r - a J =m\b — . -> -> -» -> => m r + n r =mb + n a r = mb + n a m + n Result (1): If P is the mid point of AB, then it divides AB in the ratio 1:1. .-. The P.V. of P is \ . b + \. a a + b 1 + 1 48 .-. P.V. of the mid point P of AB is OP = r = -^ — Result (2): Condition that three points may be collinear Proof: Assume that the points A, P and B (whose P.Vs are a , r and b respectively) are collinear We have -> -> — » mb + n a r = ; m + n -> -» — » (m + n) r = mb + n a (m + n) r ■Bfl =0 In this vector equation, sum of the scalar coefficients in the L.H.S. =(m + n)-m-n = Thus we have the result, if A, B, C are collinear points with position -> -> -» vectors a , b , c respectively then there exists scalars x, y, z such that -> -> -> -> jfl +yb + zc =0 and x + y + z = Conversely if the scalars x, y, z are such that x + y + z = and -» -> -» -» -> -» -> jfl +y^ +zc =0 then the points with position vectors a , £ and c are collinear. Result 3: [Section formula - External division] -> -> Let A and B be two points with position vectors a and b respectively and let P be a point dividing AB externally in the ratio m : n. Then the position vector of P is given by -» -> > m b OP = m -n a - n Proof: Let be the origin. A and B are the two points whose position ' -i /ectors are a -> and ^ Then OA = = a ; OB = Let P divide AB externally in the ratio m : n. Let the position vector of P -> > -> with respect to O be r i.e. OP = r 49 w u AP m We have pb = 7 n AP = - m PB ^> «AP = mPB ap & ft? are in the opposite direction. (cJP-Oa) = -m(oB-CJ>p) =^> n\r -~a) = m\r -~b) -» -> -> -> m & - « a = m r —nr -> — » — » -> => n r — n a = m r — mb -» -> -> => mb — n a = (m — ri) r -» -> — > mb —n a r = m - « Theorem 2.8: The medians of a triangle are concurrent. Proof: Let ABC be a triangle and let D, E, F be the mid points of its sides BC, CA and AB respectively. We have to prove that the medians AD, BE, CF are concurrent. -» -> -» Let O be the origin and a , b , c be the position vectors of A, B, C respectively. The position vectors of D, E, F are b + c c + a a + b Affl) 2 ' 2 ' 2 Let Gj be the point on AD dividing it internally in the ratio 2 : 1 B($) D C(Q F/g. 2. 75 P.V. ofGj = 2QD + 1QA 2+ 1 + c I ,-> + 1 fl OG, = V a + o + c 'l - 3 =3 Let G 2 be the point on BE dividing it internally in the ratio 2 : 1 2QE +1QB (1) OGo = 2+ 1 50 c + a + 1. b -> -:> -> a + b + c OG 2 = 3 = 3 ( 2 ) Similarly if G3 divides CF in the ratio 2 : 1 then > a + b + c OG 3 = 3 (3 ) From (1), (2), (3) we find that the position vectors of the three points Gj, G 2 , G 3 are one and the same. Hence they are not different points. Let the common point be denoted by G. Therefore the three medians are concurrent and the point of concurrence is G. Result: The point of intersection of the three medians of a triangle is called the centroid of the triangle. > a + b + c The position vector of the centroid G of A ABC is OG = 3 -» -» -> where a , b , c are the position vectors of the vertices A, B, C respectively and O is the origin of reference. -» — » — » Example 2.1: If a , b , c be the vectors represented by the three sides of a -» — » -> — » triangle, taken in order, then prove that a + b + c = O Solution: Let ABC be a triangle such that FiC = a , CA = b and AB = c a + b + c = FiC + CA + AB = BA + AB (.-. BC + CA = BA Example 2.2: If a anc hexagon, find the vectors determined by the other sides taken in order. = BB =0 If a and b are the vectors determined by two adjacent sides of a regular 51 Solution: Let ABCDEF be a regular hexagon > -> > -> such that AB = a and BC = b Since AD II BC such that AD = 2.BC AD = 2BC =Tb In AABC, we have AB + ISC = AC > -> -> => AC = a + b InAACD, AD = AC + CD Fig. 2. 17 > > — > -> /-> -A CD = AD - AC = 2 fc - V a + fe J = DE = - AB = FiF = - BC = FA= -CD =-\b - a) = Example 2.3: b — a a -^> a — b -» — » -> -> The position vectors of the points A, B, C, D are a , i , 2 a + 3o . -> -> > — > a -26 respectively. Find DB and AC Solution: Given that OA = a ; OB = b ; OC =2a + 36 ; OD = a - -> -26 DB = OB -OD = ~b - -^a -2fe^= Z? -a +2b = 36 - a AC = OC -OA /-> -A -> = \2a + 36/ - a = a + 36 Example 2.4: Find the position vector of the points which divide the join of the -» points A and B whose P.Vs are a externally in the ratio 3 : 2 2 6 and 2 a b internally and 52 Solution: OA = a -26 ; OB =2a - b Let P divide AB internally in the ratio 3 : 2 3 OB + 2QA 3(2"^ - t) + 2\a - it) P.V. ofP = 3 + 2 -> -» -» -> -» -» 6a -36 +2a -46 8a -76 8 -> 7 -> ' = c « — T b 5 5 Let Q divide AB externally in the ratio 3 : 2 ( -> -A /'-> -V ^2 g - 6 j_ n,, ^ 3QB-2QA 3^ -6^-2^ -2 , P.V. of Q = — = l -» -> -> -» -» -» = 6a -36 -2a +46 =4a +6 Example 2.5: If a and 6 are position vectors of points A and B respectively, then find the position vector of points of trisection of AB. Solution: Let P and Q be the points of trisection of AB .*"" J £ ^ /E >, Let AP = PQ = QB = X (say) ^ 2 7S P divides AB in the ratio 1:2 P.V.ofP = O? = LM±|OA = Ll±2^ = l±2l Q is the mid-point of PB -» -» -» -» -» 6 +2a -» 6 +2a +36 OP + OB 3 + * 3 2a +46 P.V. of Q = 2" -> -> g +26 3 Example 2.6: By using vectors, prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other. 53 Solution: Let ABCD be a quadrilateral q^/j c(r) First we assume that ABCD is a parallelogram To prove that its diagonals bisect each other Let O be the origin of reference. -> > -> > -> > -> .-. OA = a , OB = & , OC = c , OD = d &W B$ > > Fig. 2. 79 Since ABCD is a parallelogram AB = DC =^> OB -OA = OC -OD =^> b - a = c - d -> -» — » — » -> -» -> -> b + d a + c => o + a = a + c => ~ = ~ i.e. P.V. of the mid-point of BD = P.V. of the mid-point of AC. Thus, the point, which bisects AC also, bisects BD. Hence the diagonals of a parallelogram ABCD bisect each other. Conversely suppose that ABCD is a quadrilateral such that its diagonals bisect each other. To prove that it is a parallelogram. -> -> -> -» Let a , b , c , d be the position vectors of its vertices A, B, C and D respectively. Since diagonals AC and BD bisect each other. P.V. of the mid-point of AC = P.V. of the mid-point of BD •••(I) => -» — » -> -» a + c b + d 2 _ 2 -> -> -» -» ^>a+c = b+d => -> — » -» -» £> - a = c -a i.e. AB = DC Also(l) => -» -> -> -> d — a = c — b i.e. AD = FiC Hence ABCD is a parallelogram. •nple 2.7: In a triangle ABC if D and E are the midpoints of sides AB and AC respectively, show that BE + DC = y BC 54 Solution: For convenience we choose A as the origin. -» Let the position vectors of B and C be b and -» c respectively. Since D and E are the mid-points of AB and AC, the position vectors -> c of D and E are it and it respectively. B {$) C(c) Fig. 2. 20 Now BE = P.V. ofE-P.V. ofB = c -> DC = P.V. of C- P.V. of D = ~c -y — > > c -> -> b BE +DC =^r-6 +C 2 u -r c 2 -iC?-*) = j [P.V. of C- P.V. of B] = 2 BC Example 2.8: Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. Solution: Let ABC be a triangle, and let O be the A (a) origin of reference. Let D and E be the midpoints of AB and AC respectively. Let OA = a , OB = b , OC = c P.V. ofD=OD = P.V. of E = OE = a + 2 a + c B(b) Now DE = OE - OD = a + c a + C05 Fig. 2. 27 55 Also a + c - 2 a-fc 1 f-» = 2 V c " DE = i bc => DE II BC DE 4^ => Ide ■) 4 ( oc Ob) =\ BC DE I = o I BC I => DE = o BC 1 Hence DE II BC and DE = ^ BC. Example 2.9: Using vector method, prove that the line segments joining the mid-points of the adjacent sides of a quadrilateral taken in order form a parallelogram. Solution: Let ABCD be a quadrilateral and let P, Q, R, S be the mid-points of the sides AB, BC, CD and DA respectively. Then the position vectors of P, Q, R, S are a + b b + c c + d d + a D(d) respectively. z z z Ma) p B(b) Fig. 2. 22 In order to prove that PQRS is a parallelogram it is sufficient to show that PQ =^R and r^ = QR Now PQ = P.V. ofQ-P.V. ofP = SR = P.V. of R- P.V. of S = b + c a + b I 2 )~ V 2 J c + d d + a { 2 j v 2 J -> -> c — a c - a .-. PQ = SR =^PQ II SR and PQ = SR Similarly we can prove that PS = QR and PS II QR Hence PQRS is a parallelogram. Example 2.10 : —> -> -> Let a , b , c be the position vectors of three distinct points A, B, C. If -> — » — » there exists scalars Z, m, n (not all zero) such that Z a +m /? +n c = and Z + m + n = then show that A, B and C lie on a line. 56 Solution: It is given that I, m, n are not all zero. So, let n be a non-zero scalar. -»-»—» -> / -> -A Z a + mb + n c = => »c =—\la+mb) ( -> -*\ /-> ->\ _> _> -> ya +mb ) -> y a +mb ) la +mb c = ~ n => c = " - (/ + m) = / + m => The point C divides the line joining A and B in the ratio m : Z Hence A, B and C lies on the same line. -» — » -> -> -> -> Note : a , b are collinear vectors=> a =Xb or b =Xa for some scalar A, Collinear points: If A, B, C are three points in a plane such that AB = XBC or AB = ^AC (or) BC = XAC for some scalar X, then A, B, C are collinear. Example 2.11: Show that the points with position vectors -» — » — » -» -» -> -> — » -» a -26 +3c ,-2a +3/? - c and A a -lb +1 c are collinear. Solution: Let A, B, C be the points with position vectors -> -> — » — » — » — » -» — » — » a -2/? + 3 c , - 2 a +3/? - c and 4a -7/? + 7 c respectively. — > ->->-> — > -» -> -> — > -» ->• -> OA = a -2 b + 3c , OB =-2 a +3b - c , OC = 4 a -7ft + 7 c > > > / -> -> -A /-> -> -A AB =OB-OA =^-2a+3/?-cy-Va-2ft+3cJ -» — » — » — » — » — » -» — » — » = — 2a + 3 ft - c -a +2 b -3 c = - 3 a + 5 b -Ac > > > / -> -> -A / -> -> -A BC=OC-OB =V4a-7ft+7cJ-V-2a+3ft-c y ; -> -> — » — » — » — » -> -» — » = 4a -7ft +1 c +2 a -3ft + c = 6a -10ft + 8 c Clearly BC =6~a -10b + 8"? = - 2 {- 3~a +5~b - A~c) = -2(ab) => AB and BC are parallel vectors but B is a point common to them. So AB and BC are collinear vectors. Hence A, B, C are collinear points. 57 EXERCISE 2.1 -> -> — > — > (1) If a and b represent two adjacent sides AB and BC respectively of a paralleogram ABCD. Find the diagonals AC and BD . (2) If PO + OQ = QO + OR , show that the points P, Q, R are collinear. (3) Show that the points with position vectors -> -> — » -» — » -» -» — » a -2fc +3c ,-2a + 3 b + 2 c and- 8 a + 13 & are collinear. -> -> — » (4) Show that the points A, B, C with position vectors -2 a + 3 b + 5 c , -> — » — » -> -> a + 2 b + 3 c and 1 a - c respectively, are collinear. (5) If D is the mid-point of the side BC of a triangle ABC, prove that AB + AC =2 AD (6) If G is the centroid of a triangle ABC, prove that GA +GB +GC = O (7) If ABC and A'B'C are two triangles and G, G' be their corresponding centroids, prove that AA* + BET + CC = 3GC? (8) Prove that the sum of the vectors directed from the vertices to the mid-points of opposite sides of a triangle is zero (9) Prove by vector method that the line segment joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides and equal to half of their difference. (10) Prove by vector method that the internal bisectors of the angles of a triangle are concurrent. (11) Prove using vectors the mid-points of two opposite sides of a quadrilateral and the mid-points of the diagonals are the vertices of a parallelogram. (12) If ABCD is a quadrilateral and E and F are the mid-points of AC and BD respectively, prove that AB + AD + CB + CD = 4 EF 2.5 Resolution of a Vector Theorem 2.9 (Without Proof) : -» -> -> Let a and b be two non-collinear vectors and r be a vector coplanar -> ->->-> with them. Then r can be expressed uniquely as r = I a + mb where I, m are scalars. 58 Note : We call Z a + m b as a linear combination of vectors a and /? , where /, m are scalars. Rectangular resolution of a vector in two dimension Theorem 2.10 : If P is a point in a two dimensional plane which has coordinates (x, y) > -> -> -> -> then OP =ii + y _/' , where i and j are unit vectors along OX and OY respectively. Proof: Let P(x, y) be a point in a plane with reference to OX and OY as co-ordinate axes as shown in the figure. Draw PL perpendicular to OX. Then OL = x and LP = y -> -> Let i , j be the unit vectors along OX and OY respectively. Then OL = x i and X? =yj Y -* P(x, y) J* i\ —> i X Fig. 2. 23 Vectors OL and LP are known as the components of OP along x-axis and y-axis respectively. Now by triangle law of addition "OP = "OL + X? = x i +yj = r (say) r = x i + y y Now OP 2 = OL 2 + LP 2 = x 2 + y 2 =5> OP = V?+/ z* l7l = V* 2 + / Thus, if a point P in a plane has coordinates (x, y) then -> > -> -> (i) r = OP = x i + y j (ii) I~r1 =IopI =lx7'-l-y7 > l =Vx 2 7y 2 > -> (iii) The component of OP along x-axis is a vector x i and the — > -> component of OP along y-axis is a vector y 7 59 Components of a vector AB in terms of coordinates of A and B Let A(x 1? yj) and B(x 2 , y 2 ) be any two points in XOY plane. Let i and j be unit vectors along OX and OY respectively. AN = x 2 -Xj, BN=y 2 -y 1 .-. AN = (x 2 - jci) i , NB -> Now by triangle law of addition Y" A Bfeyi) / 7* >v A(*i,yi) O ] L M X Fjg. 2. 24 -> -» AB = AN + NB ={x 2 -x l )i + (y 2 -y{) j -> Component of AB along x-axis = (x 2 - Xj) z -» Component of AB along y-axis = (y 2 -.yi) j AB 2 = AN 2 + NB 2 = (x 2 - x,) 2 + (y 2 - y{) 2 => AB =^(x 2 -x 1 ) 2 + (y 2 -v 1 ) 2 which gives the distance between A and B. Addition, Subtraction, Multiplication of a vector by a scalar and equality of vectors in terms of components: Let a -» = aj i + a 2 j and £> = byi + b 2 j We define (i) a + £> = i^aj i + a 2 jj + \b-[ i +b 2 j) = (aj + b{) i +(a 2 + b 2 )j (ii) a — b = [a^ i +a 2 j V +^2^7 =(«i-»i) * +(a2- b 2)J (iii) ma = m ^aj i +a 2 j J =ma\ i +ma 2 j where m is a scalar a = b -» -> -> -> aj z +a 2 j = i»j j +b 2 j (iv) a = £> => aj i +a 2 j = b\ i +b 2 j => aj = £>janda 2 = £> 2 Example 2.12: Let O be the origin and P(— 2, 4) be a point in the xy-plane. > -> -> I >| Express OP in terms of vectors i and j . Also find I OP I 60 Solution: The position vector of P, OP = -2 i +4j l~OP I = l-2"f + 4"7l = \j(- 2) 2 + (4) 2 =^4+16 =a/20 = 2^5 Example 2.13: Find the components along the coordinates of the position vector of P(- 4, 3) Solution: > -> -> The position vector of P = OP = - 4 i + 3 j > -> Component of OP along x-axis is - 4 i i.e. component of OP along x-axis is a vector of magnitude 4 and its direction is along the negative direction of x-axis. > -> Component of OP along j-axis is 3 j i.e. the component of OP along y -axis is a vector of magnitude 3, having its direction along the positive direction of y-axis. > -> -> Example 2.14: Express AB in terms of unit vectors i and j , where the points are A(- 6, 3) and B(- 2, - 5). Find also I AB I Solution: Given OA = - 6 i + 3 j ; OB = - 2 i -5j :. AB = OB -OA = {-2~t-5~f) -(-6~f + 3~f) -> -> = 4i -8y IabI = U"f-87l = V(4) 2 + (- 8) 2 =^16 + 64 =A /80 = 4^5 Theorem 2.11 (Without Proof) : -» -> -» -> If a , /? , c are three given non-coplanar vectors then every vector r in -» -> -> -» space can be uniquely expressed as r = Z a + m b + « c for some scalars Z, m and n 61 Rectangular Resolution of a vector in three dimension Theorem 2.12: are unit vectors If a point P in space has coordinate (x, y, z) then its position vector r is ->->-> |-»| Hi 5 2 ->->-> xi + y y + z & and I r I = "\/;c + y + z where i , j , k along OX, OY and OZ respectively. Proof: OX, OY, OZ are three mutually -» -» -» perpendicular axes, i , y , & are unit vectors along OX, OY, OZ respectively. Let P be any point (x, y, z) in space and let OP =~r Draw PQ perpendicular to XOY plane and QR perpendicular to OX Then OR = x ; RQ = y ; QP = z :. OR = x i ; RQ =yj; ^P = z k U V. T V / / ' ",'*' s ''•.90° A Q X Fig. 2. 25 Now OP = OQ + QP = OR + RQ + QP — > -» -> -> OP = x j + y y + z & -> -» -> -> r = jc i +yy + z A: -» Thus if P is a point (x, y, z) and r is the position vector of P, then -> -» -» -> r =ii + y _/' + z & From the right angled triangle OQP, OP 2 =OQ 2 + QP 2 From the right angled triangle ORQ, OQ 2 =OR 2 + RQ 2 .-. OP 2 = OR 2 + RQ 2 + QP 2 OP 2 = x 2 + y 2 + z 2 OP = -\jx 2 + y 2 + z 2 => r = V?+ HI ./ 2^ 2^ 2 . . r = I r I = ^/x +y + z 2 , 2 y + z 2.6 Direction cosines and direction ratios Let P(x, y, z) be any point in space with reference to a rectangular coordinate system O (XYZ). Let a, (3 and y be the angles made by OP with the positive direction of coordinate axes OX, OY, OZ respectively. Then cosa, cos(3, cosy are called the direction cosines of OP . 62 In the fig 2.25 |OQP = 90° ; | POZ = y :. \ OPQ = y (QPIIOZ) PQ z x y :. cosy = Qp => cosy = ~ Similarly cosa = ~ and cos(3 = ~; > x y z /" ? 2 2 .". The direction cosines of OP are ~ ~, ~ where r = ^\Jx +y + z Result 1: Sum of the squares of direction cosines is unity. 2 2o 2 (A 2 (jV fz\2 x 2 + y 2 + Z 2 cos^a + cos z p + cos z y = \jj + (£J + \^J = \ 2 = % = 1 [v r 2 = x 2 + / + z 2 ] r 2 2 2 .". cos a + cos (3 + cos y = 1 Result 2: Sum of the squares of direction sines is 2. 2 2 2 2 2 2 sin a + sin p + sin y = (1 - cos a) + (1 - cos p) + (1 - cos y) 2 2 2 = 3 - [cos a + cos P + cos y] = 3-1 =2 2 2 2 sin a + sin P + sin y = 2 Direction ratios: Any three numbers proportional to direction cosines of a vector are called its direction ratios, (d. r's). -> -» -> -» Let r =xi + y j + zk be any vector - > x y z f 2 2 2 => Direction cosines of r are ~ ~;, ~ where r = AJx + y + z x y z => cos a = ~ ; cos p = ~. ; cos y = ~ where a, P, y be the angles made -» by r with the coordinate axes OX, OY, OZ respectively ^ x y osa i _L_ __21 — r — — — r — r cosa ' cosp ' cosy cosa cosp cosy => x : y : z = cosa : cosp : cosy i.e. the coefficients of i, j, k in the rectangular resolution of a vector are proportional to the direction cosines of that vector. -> -> -> -> .'. x, y, z are the direction ratios of the vector r =x i +y I + z k 63 Addition, Subtraction and Multiplication of a vector by a scalar and equality in terms of components: -> -»—»-» -» -> — » -> Let a = a j i + a 2 j + a^k and £> = b± i + b 2 j + 03 k be any two vectors. Then -> -» -> -> -> (i) a + o = (fli + Pi) ' + ( a 2 + ^2)J + ( a 3 + 3) ' c -» -» -> .— » .-> (ii) a —b = (a^-bAi +(^2~^2)J + ( a 3~^2) k -> f -> -> -> (iii) ma = m^a ^ i + a 2 j + a^k -> -> -» = ma j i + ma 2 7 + »«*3 A; where m is a scalar -> -> (iv) a = b <=> a\ = b\, a 2 = b 2 and 03 = 03 Distance between two points: Let A (jcj, yi, Zi) and B(x 2 , y 2 , z 2 ) be any two points Then AB = OB-OA -> -> -A f -» -> -> *2 * + ^2 7 + ^2 * y - \ x i i +y\j +zi k -» -> -> = (x 2 -x 1 ) 1 +(y 2 -yi)j +(z 2 -zi) k :. The distance between A and B is AB = I AB I ABl = -» -> -> (*2 - jc,) i + (y 2 - y,) 7 + (z 2 - zj) fc = V(^2 - *i) 2 + (yi - y\) 2 + (Z2 - z i) 2 -» — » — » Example 2.15: Find the magnitude and direction cosines of 2 i -7 + 7 fc Solution: Magnitude of it- ~J + l~t = \l~t - ~j + lt\ = aJ(2) 2 + (- l) 2 + (7) 2 = ^4+1+49 =-^54 =3yf6 -> -> -> 2 17 Direction cosines of 2 i -7 + 7 fc are t - 7= ' - t - 7= ' t-7=t Example 2.16: Find the unit vector in the direction of 3 i + 4 j - 12 k 64 Solution: Let a = 3 j + 4 7 - 12 k \t\ = I3~t + 4~J - I2~k\= V(3) 2 + (4) 2 + (- 12) 2 = a/9 + 16 + 144 =Vl69 =13 -» -» -» -> .,_,.. ,-> . a a 3 i + 4 ./ - 12 k Unit vector in the direction or a is a = ~ — ~ = f5 \?\ -> -> -> -» — » — » Example 2.17: Find the sum of the vectors j -7 + 2 & and 2 i + 3j -4 k and also find the modulus of the sum. Solution: -> -> -> -> -> -> -> -> Let a = i — j +2k , b =2 i +3 j -4 k -> -> /-> -> -»\ / -> -> -A -> -> -> a + £ =^j-;'+2fe^+V,2j+3;-4Jfe^=3j+2;'-2A: l~a+in = ^3 2 + 2 2 + (- 2) 2 = ^9 + 4 + 4 = -y/l7 Example 2.18: If the position vectors of the two points A and B are i +2 j -3 k and 2 i -4 j + k respectively then find I AB I Solution: If O be the origin, then OA = i + 2 7 -3 k, OB =2 i - 4 7 + k AB = OB - OA = (2~t-4? + ~lt) - (~t + 2~f - 3~k) -> -> -> = i - 6 7 + 4 jfe IabI = V(l) 2 + (- 6) 2 + (4) 2 = a/53 Example 2.19: Find the unit vectors parallel to the vector - 3 i + 4 7 Solution: Let a = - 3 i + 4 j \t\ = yj(- 3) 2 + 4 2 = a/9 +16 =5 65 a a 1 -> 1 ( -> -> -» a f- 3 -> 4 -> Unit vectors parallel to a are + a = ± \—r~ i +~F j Example 2.20: Find the vectors of magnitude 5 units, which are parallel to the vector 2i - j -> -» -> Solution: Let a = 2 i - j \7\=yJ2 2 + (-l) 2 =^5 a a 1 / -> -A 2 -» 1 -> -> -> A Vectors of magnitude 5 parallel to 2 i - _/' = ± 5 a -» — » -> Example 2.21: Show that the points whose position vectors 2 i + 3 j -5k, -> -> -> -> — » — » 3 i + _/' -2J and 6 i - 5 _/' + 7 & are collinear. Solution: Let the points be A, B and C and O be the origin. Then > ->->-> — > ->->-> > -> -> — » OA = 2 i + 3 y - 5 jfe ; OB = 3 j + j - 2 jfc ; OC = 6 i - 5 j + 7 k — > — > — > / -> -> -A / -> -> -A .-. AB = OB - OA = ^3 i + j - 2 k ) - \2 i + 3j -5 k ) -» -» -> = i -2j +3k AC = OC - OA =\6i -5j +1 k) -\2i + 3j -5 k ) AC = 4 j - 8 j + 12 k = 4 V i - 2 j + 3 k ) = 4 AB Hence the vectors AB and AC are parallel. Further they have the common point A. .". The points A, B, C are collinear. 66 Example 2.22: If the position vectors of A and B are 3 i - 7 j -Ik and ->->->■ — > 5 z +4 j +3 k , find AB and determine its magnitude and direction cosines. Solution: Let O be the origin. Then > -» -> -» > -> -> -» OA = 3 j - 7 y - 7 Jfc , OB =5 i + 4j +3k AB = OB - OA = ^5 i + 4 y + 3 k ) - \3 i - 7 j - 7 k ) AB = 2 j + 11 j + 10 fc I AB I = a/(2) 2 + (H) 2 + (10) 2 = 15 m ,. . • 2 11 10 The direction cosines are tt , tf , tt EXERCISE 2.2 (1) Find the sum of the vectors 4 i +5j + k,-2i + 4j - k and -> -» -» 3 i - 4 y + 5 /c . Find also the magnitude of the sum. -» — » — » — » — » -» — » — » -» — » — » — » (2) If a = 3 i - j -4 k, b = - 2 i + 4 j - 3 k and c = i + 2 j - k find l2a - fc +3c I (3) The position vectors of the vertices A, B, C of a triangle ABC are respectively -> -> -» — » — » -» -» — » -> 2 i + 3 7 + 4 k , - i +2 j - k and 3 i -5 j +6 k Find the vectors determined by the sides and calculate the length of the sides. (4) Show that the points whose position vectors given by -> -> -» -> -> -> — » — » (i) - 2 i + 3 j +5 k , i +2j + 3 k , 1 i - k -» — » — » -> -» — » -> -> (ii) z -2 j +3fe, 2 i +3y -4/c and -7 7 + 10 /c are collinear. -» — » — » -> -» — » (5) If the vectors a =2i -3; and b = — 6 i +m; are collinear, find the value of m. -> r- -> (6) Find a unit vector in the direction of i + ^J3 j 67 -> -> -» — » -> -» -> -> -> 4 i + 5 7 + 6 fe , 5 i + 6 j + 4 k , 6 i + 4 j + 5 k . Prove that the (7) Find the unit vectors parallel to the sum of 3 i - 5 j + 8 k -> -> and - 2 _/' - 2 & -> -» -» -» -» -» -» (8) Find the unit vectors parallel to 3a - 2 fc + 4 c where a=3 i - j -4 k, -> -» -» -» -» -» -» -» &=-2f+4y'-3fe, c = i + 2 j - k (9) The vertices of a triangle have position vectors -> -> -> 4i + 5 7 + 6 fc , 5 triangle is equilateral. -» -» -» -» -> -> -> -> -> (10) Show that the vectors 2 i - j + k , 3 i - 4 7 - 4 k , i -3j -5 k form a right angled triangle. (11) Prove that the points 2 i + 3 j +4k, 3 i +4j +2k, 4 i +2 j +3 k form an equilateral triangle. -» -» -» (12) If the vertices of a triangle have position vectors i +2y +3fc, -» -» -» -» -» — » 2 j + 3 j + k and 3 i + j + 2 k , find the position vector of its centroid. (13) If the position vectors of P and Q are i +3 j -Ik -> -> -> > and 5 i - 2 7 + 4 k , find PQ and determine its direction cosines. (14) Show that the following vectors are coplanar -> — » — » -» — » -> — » — » (i) j - 2 y + 3 k , - 2 i +3 j -4 k , - j +2k -» — » -» — » — » — » — » -» — » (ii) 5i + 6 j + 7 jfe , 7 i -87 +9k, 3 i +20 7 + 5 it -> -» -» -» -> (15) Show that the points given by the vectors 4 i +5 7 + k,— j - k, -» -» -» -» -> -» 3 j +9y + 4 & and - 4 i + 4 7 + 4 & are coplanar. -> -> -> -» -» -» (16) Examine whether the vectors i +3 j + k , 2 i - j - k -> -> and 77 + 5 fc are coplanar. 68 3. ALGEBRA 3.1 Partial Fractions: Definitions: Rational Expression: An expression of the form ~r^- where p(x) and q(x) * are polynomials in x is called a rational expression. 2 5x -2 3x +2x-\ The expressions ~2 , ~~2 are examples for rational x +3x + 2 x + x - 22 expressions. Proper Fraction: A proper fraction is one in which the degree of the numerator is less than the degree of the denominator. 3 X+ 1 Ix + 9 The expressions 2 > ~~3 2 are examples for proper x + 4x +3 x + x - 5 fractions. Improper Fraction: An improper fraction is a fraction in which the degree of the numerator is greater than or equal to the degree of the denominator. 3 c 2 a 2 x + 5x + 4 x — x + 1 The expressions ~~ 2 , ~2 are examples for improper x +2x + 3 x +x + 3 fractions. Partial Fraction: 7 Consider the sum of ^ and x — 2 x- 1 We simplify it as follows: 7 5 l(x - 1) + 5(x - 2) 7x-7 + 5x-10 12* -17 jc-2 + x-l _ (jc-2)(jc-1) ~ (*-2)(x-l) ~(x-2)(x-l) 12x- 17 Conversely the process of writing the given fraction , y\ (x - V\ as 7 5 2 + x _ i 1S known as splitting into partial fractions (or) expressing as partial fractions. A given proper fraction can be expressed as the sum of other simple fractions corresponding to the factors of the denominator of the given proper fraction. This process is called splitting into Partial Fractions. If the given d(x\ fraction ~r^- is improper then convert into sum of a polynomial expression and a proper rational fraction by dividing p(x) by q(x). 69 Working Rule : Given the proper fraction ~r^- . Factorise q(x) into prime factors. Type 1: Linear factors, none of which is repeated. If a linear factor ax + b is a factor of the denominator q(x) then corresponding to this factor associate a simple fraction , , where A is a constant (A ^ 0). i.e., When the factors of the denominator of the given fraction are all linear factors none of which is repeated, we write the partial fraction as follows : x + 3 A B 7 — , S \ I", — TIT = — TT + t — TT where A and B are constants to (x + 5)(2x+l) x + 5 2x+\ be determined. 3x + 7 Example 3.1: Resolve into partial fractions -j x - 3x + 2 2 The denominator x - 3x + 2 can be factorised into linear factors. 2 2 x —3x + 2=x — x — 2x + 2 = x (x — 1) - 2 (x - 1) = (x - 1) (x — 2) 3x + l A B We assume -7 = r + T where A and B are constants. jc -3jc + 2 x ~ l x ~ 2 3x + 7 A(x-2) + B(x-l) ^ x 2 -3x + 2 " (x-l)(*-2) ^ 3x + 7 = A(x-2) + B(x-l) ...(1) Equating the coefficients of like powers of x, we get Coefficient of x : A + B = 3 ... (2) Constant term : - 2A - B = 7 ... (3) Solving (2) and (3) we get A = -10 B = 13 3x + 7 -10 13 13 10 x 3x + 2 x — 1 x — 2 x — 2 x — 1 Note: The constants A and B can also be found by successively giving suitable values for x. To find A, put x = 1 in (1) 3(1) + 7 = A(l-2) + B(0) 10 = A(-l) A = -10 70 To find B, put x = 2 in (1) 3(2) + 7 = A(0) + B(2 - 1) B = 13 3x + 7 -10 13 x l - 3x + 2 ~x-l x-2 3x + 7 13 10 ~7T x + 4 3x + 2 ^ - 2 x - 1 Example: 3.2: Resolve into partial fractions — r (x - 4) (x + 1) 2 The denominator (x - 4) (x + 1) can be further factored into linear factors i.e. (x 2 - 4) (x + 1) = (x + 2) (x -2)(x+ 1) x + 4 ABC, Let — 7 = — ~z + + — — r, where A, B and C are (x -4)(x+l) x + 2 x-2 x+l' constants to be determined. x + 4 _ A(x - 2) (x + 1) + B(x + 2) (x + 1) + C(x + 2) (x - 2) (x 2 -4)(x+l) ~ (x + 2) (x - 2) (x + 1) => x + 4 = A(x-2)(x+l) + B(x + 2)(x+l) + C(x + 2)(x-2) ...(1) To find A, put x = - 2 in (1) -2 + 4 = A(-2-2)(-2+l) + B(0) + C(0) 2 = 4A => A = 1/2 To find B, put x = 2 in (1), we get B = 1/2 To find C, put x = - 1 in (1), we get C = - 1 • x + 4 _ 1/2 (x 2 -4)(x+l) ~ ( X+T > _ x + 4 __ 1 ^ (x 2 -4)(x+l) " 2 ^ + 2 ) + 2(x-2) "x+l Type 2: Linear factors, some of which are repeated If a linear factor ax + b occurs n times as a factor of the denominator of the given fraction, then corresponding to these factors associate the sum of n simple fractions, Ai A 2 A3 A n 1, 'Z (- 1; (x- -2) """x+l 1 1 + + o +...+' ax + b (ax + b) 2 (ax + bf ■'■ (ax + bf Where A1 , A2, A3, ... A„ are constants. 71 9 xumple 3.3: Resolve into partial fi actions ') (x - 1) (x + 2) 9 ABC Lei •> — , + , i + ') (x-l)(x + 2) x ~ l x + 1 (x + 2) 9 A(x + 2) 2 + B(x - 1) (x + 2) + C(x - -1) (x - 1) (x + 2) 2 ~ (x - 1) (x + 2) 2 => 9 = A(x + 2) 2 + B(x - 1) (x + 2) + C(x - -1) To find A, put x = 1 in (1) We get 9 = A (1 + 2) 2 => A = 1 To find C, put x = -2 in (1) We get 9 = C (- 2 - 1) => C = - 3 In (1), equating the coefficient of x on both sides we get A + B = => 1+B = =>B = -1 9 113 " (x - 1) (x + 2) Z ~ x - 1 x + 2 ( x + 2) 2 (1) Type 3: Quadratic factors, none of which is repeated 2 If a quadratic factor ax + bx + c which is not factorable into linear factors occurs only once as a factor of the denominator of the given fraction, then Ax+B corresponding to this factor associate a partial fraction — 2~^ where A ax + bx + c and B are constants which are not both zeros. Consider 1 (x + 1) (x + 1) • , • £■■-,* 2x A Bx + C We can write this proper fraction in the form 2 = I i +~ 2 (x+ 1) (x + 1) x+ 1 x+1 The first factor of the denominator x + 1 is of first degree, so we assume its numerator as a constant A. The second factor of the denominator x +1 is of 2 degree and which is not factorable into linear factors. We assume its numerator as a general first-degree expression Bx + C. x - 2x - 9 Example 3.4: Resolve into partial fractions — 2 (x + x + 6) (x + 1) 72 2 x -2x-9 Ax + B C Let —2 = ~2 + v 7 i (x +x + 6)(x+l) x +x + 6 x+1 x 2 -2x-9 (Ax + B)(x+ 1) + C(x 2 + x + 6) => ~ 2 = 2 (x +x + 6) (x+ 1) (x +x + 6) (x + 1) => x 2 -2x-9 = (Ax+B)(x+1) +C(x 2 + x + 6) ... (1) To find C put x = — 1 in (1) We get l+2-9 = C(l-l+6)=>C=-l To find B, put x = in (1) We get - 9 = B + 6C -9 = B-6 => B = -3 To find A, Put jc = 1 in (1) 1 - 2 - 9 = (A - 3) (2) + (- 1) (8) => - 10 = 2A - 14 A = 2 * -2x-9 2x - 3 1 ~2 ;tt — r: =t (x + x + 6)(x+l) x +x + 6 x+ ^ 2 x + x + 1 Example 3.5: Resolve into partial fractions ~2 x - 5x + 6 Solution: Here the degree of the numerator is same as the degree of the denominator, i.e. an improper fraction. x On division ~2 — ~ 7 = 1 + 2" \ •■■(!) Let x - 6x - 5 = A(x - 3) + B(x - 2) By putting x = 2, -A=12-5 =^> A = - 7 By putting x = 3, B = 18-5 =^ B = 13 2 x +x + 1 6x-5 - 1 +T x - 5x + 6 A B x - 2 x - 3 x - 5x + 6 6x- 5 r — rijc + n • (1) 2 X + X+ 1 = 1 7 x - X 2 7 13 2 x - 2 X -5x + 6 + x+ 1 X- 3 13 2 x ■ -5x + 6 2 ~^x-3 73 7x-l (i) 6 - 5x + x x-2 (x + 2) (jc - l) 2 2 2x - 5x — 7 (x-2) 3 2 7x - 25* + 6 2 x + x + 1 (JC- (6) (9) ■ (12; 1)(JC -2)(x- x+ 1 ■3) (X- - 2) 2 (x + x 2 -3 3) (x + 2) (x + 2 X +x + 1 1) • 2 EXERCISE 3.1 Resolve into partial fractions (1) (x-lAx+l) ™ (4) : n ) m1 .i (5) (x- 1) (x + 2) 2 x - 6x + 2 (7) ^TT^T (8) x (x + 2) x + 2 (10) 2 (11)— 2" (x + 1) (x + 1) (x - 2x - 1) (3x - 2) x + 2x +1 3.2 Permutations: Factorial: The continued product of first w natural numbers is called the "n factorial" and is denoted by n ! or \n_ i.e. n\ = 1x2x3x4x...x(m-1)xm 5! = 1x2x3x4x5= 120 Zero Factorial: We will require zero factorial in the latter sections of this chapter. It does not make any sense to define it as the product of the integers from 1 to zero. So, we define ! = 1 . Deduction: n\ = Ix2x3x4x...x(w-l)xw = [Ix2x3x4x...x(n- l)]n = [(«-l)!]« Thus, n\ = n [(n - 1)!] For example, 8 ! = 8(7 !) 3.2.1 Fundamental Principles of Counting: In this section we shall discuss two fundamental principles viz., principle of addition and principle of multiplication. These two principles will enable us to understand permutations and combinations and form the base for permutations and combinations. 74 Fundamental Principle of Multiplication: If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any one of these m ways, second job can be completed in n ways; then the two jobs in succession can be completed in m x n ways. Explanation: If the first job is performed in any one of the m ways, we can associate with this any one of the n ways of performing the second job: and thus there are n ways of performing the two jobs without considering more than one way of performing the first; and so corresponding to each of the m ways of performing the first job, we have n ways of performing the second job. Hence, the number of ways in which the two jobs can be performed is m x n. Example 3.6: In a class there are 15 boys and 20 girls. The teacher wants to select a boy and a girl to represent the class in a function. In how many ways can the teacher make this selection? Solution : Here the teacher is to perform two jobs : (i) Selecting a boy among 15 boys, and (ii) Selecting a girl among 20 girls The first of these can be performed in 15 ways and the second in 20 ways. Therefore by the fundamental principle of multiplication, the required number of ways is 15 x 20 = 300. Fundamental Principle of Addition: If there are two jobs such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m + n) ways. Example 3.7: In a class there are 20 boys and 10 girls. The teacher wants to select either a boy or a girl to represent the class in a function. In how many ways can the teacher make this selection? Solution: Here the teacher is to perform either of the following two jobs : (i) selecting a boy among 20 boys, (or) (ii) Selecting a girl among 10 girls The first of these can be performed in 20 ways and the second in 10 ways. Therefore, by fundamental principle of addition either of the two jobs can be performed in (20 + 10) = 30 ways. 75 Hence, the teacher can make the selection of either a boy or a girl in 30 ways. Example 3.8: A room has 10 doors. In how many ways can a man enter the room through one door and come out through a different door? Solution: Clearly, a person can enter the room through any one of the ten doors. So, there are ten ways of entering into the room. After entering into the room, the man can come out through any one of the remaining 9 doors. So, he can come out through a different door in 9 ways. Hence, the number of ways in which a man can enter a room through one door and come out through a different door = 10 x 9 = 90. Example 3.9: How many words (with or without meaning) of three distinct letters of the English alphabets are there? Solution: Here we have to fill up three places by distinct letters of the English alphabets. Since there are 26 letters of the English alphabet, the first place can be filled by any of these letters. So, there are 26 ways of filling up the first place. Now, the second place can be filled up by any of the remaining 25 letters. So, there are 25 ways of filling up the second place. After filling up the first two places only 24 letters are left to fill up the third place. So, the third place can be filled in 24 ways. Hence, the required number of words = 26 x 25 x 24 = 15600 Example 3.10: How many three-digit numbers can be formed by using the digits 1, 2, 3, 4, 5. Solution : We have to determine the total number of three digit numbers formed by using the digits 1, 2, 3, 4, 5. Clearly, the repetition of digits is allowed. A three digit number has three places viz. unit's, ten's and hundred's. Unit's place can be filled by any of the digits 1, 2, 3, 4, 5. So unit's place can be filled in 5 ways. Similarly, each one of the ten's and hundred's place can be filled in 5 ways. .•. Total number of required numbers = 5 x 5 x 5 = 125 76 Example 3.11: There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 5 each? Solution: Here we have to perform 6 jobs of answering 6 multiple choice questions. Each of the first three questions can be answered in 4 ways and each one of the next three can be answered in 5 different ways. So, the total number of different sequences = 4x4x4x5x5x5 = 8000 Example 3.12: How many three-digit numbers greater than 600 can be formed by using the digits 4, 5, 6, 7, 8? Solution: Clearly, repetition of digits is allowed. Since a three-digit number greater than 600 will have 6, 7 or 8 at hundred's place. So, hundred's place can be filled in 3 ways. Each of the ten's and one's place can be filled in 5 ways. Hence, total number of required numbers = 3x5x5 = 75 Example 3.13: How many numbers divisible by 5 and lying between 5000 and 6000 can be formed from the digits 5, 6, 7, 8 and 9? Solution: Clearly, a number between 5000 and 6000 must have 5 at thousand's place. Since the number is divisible by 5 it must have 5 at unit' s place. Now, each of the remaining places (viz. Hundred's and ten's) can be filled in 5 ways. Hence the total number of required numbers = 1x5x5x1= 25 Example 3.14: How many three digit odd numbers can be formed by using the digits 4, 5, 6, 7, 8, 9 if : (i) the repetition of digits is not allowed? (ii) the repetition of digits is allowed? Solution: For a number to be odd, we must have 5, 7 or 9 at the unit's place. So, there are 3 ways of filling the unit's place. (i) Since the repetition of digits is not allowed, the ten's place can be filled with any of the remaining 5 digits in 5 ways. Now, four digits are left. So, hundred's place can be filled in 4 ways. 77 So, required number of numbers = 3x5x4 = 60 (ii) Since the repetition of digits is allowed, so each of the ten's and hundred's place can be filled in 6 ways. Hence required number of numbers = 3x6x6=1 08 EXERCISE 3.2 1. In a class there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent a competition. In how many ways can the teacher make this selection? 2. Given 7 flags of different colours, how many different signals can be generated if a signal requires the use of two flags, one below the other? 3. A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties and 5 pencil varieties, in how many ways can he select these articles? 4. Twelve students compete in a race. In how many ways first three prizes be given? 5. From among the 36 teachers in a college, one principal, one vice- principal and the teacher-in charge are to be appointed. In how many ways this can be done? 6. There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 2 each? 7. How many numbers are there between 500 and 1000 which have exactly one of their digits as 8? 8. How many five-digit number license plates can be made if (i) first digit cannot be zero and the repetition of digits is not allowed. (ii) the first digit cannot be zero, but the repetition of digits is allowed? 9. How many different numbers of six digits can be formed from the digits 2, 3, 0, 7, 9, 5 when repetition of digits is not allowed? 10. How many odd numbers less than 1000 can be formed by using the digits 0, 3, 5, 7 when repetition of digits is not allowed? 11. In how many ways can an examinee answer a set of 5 true / false type questions? 78 12. How many 4-digit numbers are there? 13. How many three - letter words can be formed using a, b, c, d, e if : (i) repetition is allowed (ii) repetition is not allowed? 14. A coin is tossed five times and outcomes are recorded. How many possible outcomes are there? 3.2.2. Concept of Permutations: The word permutation means arrangement. For example, given 3 letters a, b, c suppose we arrange them taking 2 at a time. The various arrangements are ab, ba, be, cb, ac, ca. Hence the number of arrangements of 3 things taken 2 at a time is 6 and this can be written as 3P2 = 6. Definition: The number of arrangements that can be made out of n things taking r at a time is called the number of permutations of n things taken r at a time. Notation: If n and r are positive integers such that 1< r < n, then the number of all permutations of n distinct things, taken r at a time is denoted by the symbol P(w, r) or nPr. We use the symbol nVr throughout our discussion. Thus nPr = Total number of permutations of n distinct things taken r at a time. Note: In permutations the order of arrangement is taken into account; when the order is changed, a different permutation is obtained. Example 3.15: Write down all the permutations of the vowels A, E, I, O, U in English alphabets taking 3 at a time and starting with E. Solution: The permutations of vowels A, E, I, O, U taking three at a time and starting with E are EAI, EIA, EIO, EOI, EOU, EUO, EAO, EOA, EIU, EUI, EAU, EUA. Clearly there are 12 permutations. Theorem 3.1: Let r and n be positive integers such that 1 < r < n. Then the number of all permutations of n distinct things taken r at a time is given by n(n - 1) (n — 2). . . \n - r -l) i.e. nPr = n(n — 1) (n — 2) ... (n — r -l) 79 Proof: Let nPr denote the number of permutations of n things taken r at a time. Clearly the total number of permutations required is same as the number of possible ways of filling up r blank spaces by n things. □ □□•■•□ 12 3 r Let there be r blank spaces arranged in a row The first place can be filled by any one of the n things in n ways. If the first place is filled up by any one of the n things, there will be (n - 1) things remaining. Now the second place can be filled up by any one of the (n - 1) remaining things. Here it can be filled up in (n - 1) ways. Hence the first two places can be together filled in n(n - 1) ways. Having filled up these two places, we have (n - 2) things remaining with which we can fill up the third place. So the third place can be filled up by any one of these things in (n - 2) ways. Hence the first three places can be together filled in n(n - 1) (n — 2) ways. Proceeding in this way, we find that the total number of ways of filling up the r spaces is n(n - 1) (n — 2). . . upto r factors i.e. n(n - I) (n-2) ... \n — r—l) .". nPr = n(n - Y) (n-2) ... \n- r -\) = n(n - 1) (n — 2) ... (n- r + 1) Theorem 3.2: n\ Let r and n be positive integers such that 1 < r < n. Then nVr = — r ° (n-r)\ Proof: riPr = n(n - I) (n-2) ... (n- r -l) n(n - 1) (n - 2)... (n - r-l) (n -r)(n- r"+l)...2.1 (n-r) (n-r+l) ... 2.1 n\ ~ (n-r)\ 80 Theorem 3.3: The number of all permutations of n distinct things, taken all at a time is n\ Proof: nPr = n(n- 1) (n-2) ... \n- r-l) By putting r = n, nPn = n(n - I) (n-2) ... (n- n -l) = n(n - 1) (n - 2) ...(«- n -l) = n(n-l)(n-2) ... 1 = n\ :. riPn = n\ Remark: We have already defined 0! = 1. This can also be derived as follows. We know that By putting r = n, P n] nT '-(n-r)\ nVn - , ,, (n — n)\ => n\ n\ = Tjj (v nPn = n\) => n\ 0! = ^y = 1 0! = 1 Example 3.16: Evaluate 8P 3 Solution: 8! 8! (8x7x6)x5! aVi "(8 -3)! _ 5! - 5! =8x7x6 = 336 Example 3.17 : If 5Pr =6P r _i, find r Solution: 5Pr = 6P r _i 5! 6! (5-r)! ( 6 _-i), 5! 6x5! (5-r)! - (7-r)! 5! 6x5! (5-r)! {(7-r)(6-r)}(5-r)! 1= 6 - (7-r)(6-r) => (7-r)(6-r) = 6 => 42 -7r-6r+ ?-6 = => r 2 -13r+36 = => (r-9)(r-4) = => r=9orr=4 => r = 4 ( v 5Pr is meaningful for r < 5) Example 3.18: If WP4 = 360, find the value of n. n ! Solution: uPa = 360 => ; 777 =6x5x4x3 (m - 4) ! n ! 6x5x4x3x2! 6! -" (n-4)! " 2! - 2! => n! = 6! ^> « = 6 Example 3.19: If 9Pr = 3024, find r. Solution: 9Pr = 3024 =^> =9x8x7x6 = 9P 4 ^> r = 4 Example 3.20: If(n-l)P3 :«P4 = 1 :9, find «. Solution: (n - 1)P3 : MP4 = 1 : 9 => (n - 1) (n - 2) (« - 3) : n(n - 1) (n - 2) (n - 3) = 1 : 9 => i.e. 9(m - 1) (n - 2) (n - 3) = «(« - 1) (n - 2) (n - 3) ^« = 9 Example 3.21: In how many ways can five children stand in a queue? Solution: The number of ways in which 5 persons can stand in a queue is same as the number of arrangements of 5 different things taken all at a time. Hence the required number of ways = 5 P5 = 5! =120 Example 3.22: How many different signals can be made by hoisting 6 differently coloured flags one above the other, when any number of them may be hoisted at one time? 82 Solution: The signals can be made by using at a time one or two or three or four or five or six flags. The total number of signals when r-flags are used at a time from 6 flags is equal to the number of arrangements of 6, taking r at a time i.e. 6P r Hence, by the fundamental principle of addition, the total number of different signals = 6Pl + 6P2 + 6P3 + 6P4 + 6P5 + 6P6 = 6 + (6x5) + (6x5x4) + (6 x5x4x 3) + (6x5x4x3 x2) + (6x5x4x3x2x1) = 6 + 30 + 120 + 360 + 720 + 720 = 1956 Example 3.23: Find the number of different 4-letter words with or without meanings, that can be formed from the letters of the word 'NUMBER' Solution: There are 6 letters in the word 'NUMBER' . So, the number of 4-letter words = the number of arrangements of 6 letters taken 4 at a time = 6P4 = 360 Example 3.24: A family of 4 brothers and 3 sisters is to be arranged in a row, for a photograph. In how many ways can they be seated, if (i) all the sisters sit together. (ii) all the sisters are not together. Solution : (i) Since the 3 sisters are inseparable, consider them as one single unit. This together with the 4 brothers make 5 persons who can be arranged among themselves in 5! ways. In everyone of these permutations, the 3 sisters can be rearranged among themselves in 3! ways. Hence the total number of arrangements required = 5! x 3! = 120 x 6 = 720 (ii) The number of arrangements of all the 7 persons without any restriction =7! =5040 Number of arrangements in which all the sisters sit together = 720 .". Number of arrangements required = 5040 - 720 = 4320 83 3.2.3 Permutations of objects not all distinct: The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p + q = n, . n\ is — ; — r p\q\ Example 3.25: How many permutations of the letters of the word 'APPLE' are there? Solution: Here there are 5 letters, two of which are of the same kind. The others are each of its own kind. 5! 5! 120 .". Required number of permutations is = tmTTTTT = ?7 = ~ 1 ?~ = ^ Example 3.26: How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places? Solution: There are 4 odd digits 1, 1, 3, 3 and 4 odd places. 4! So odd digits can be arranged in odd places in ~, ~, ways. 3! The remaining 3 even digits 2, 2, 4 can be arranged in 3 even places in ~7 ways. 4! 3! Hence, the required number of numbers = 9T2T X V =6x3 = 18 Example 3.27: How many arrangements can be made with the letters of the word "MATHEMATICS"? Solution: There are 11 letters in the word 'MATHEMATICS' of which two are M's, two are A's, two are T's and all other are distinct. 11! .•. required number of arrangements = — — — — — = 4989600 3.2.4 Permutations when objects can repeat: The number of permutations of n different things, taken r at a time, when f each may be repeated any number of times in each arrangement, is n Consider the following example: In how many ways can 2 different balls be distributed among 3 boxes? Let A and B be the 2 balls. The different ways are 84 ox 1 Box 2 Box 3 A B D m A D n A U □ m A s n B s n A D D |AB |AB| D D □ |AB| D i.e. 9 ways. By formula n r = 3 =9 ways Example 3.28: In how many ways can 5 different balls be distributed among 3 boxes? Solution: There are 5 balls and each ball can be placed in 3 ways. So the total number of ways = 3 = 243 Example: 3.29: In how many ways can 3 prizes be distributed among 4 boys, when (i) no boy gets more than one prize? (ii) a boy may get any number of prizes? (iii) no boy gets all the prizes? Solution: (i) The total number of ways is the number of arrangements of 4 taken 3 at a time. So, the required number of ways = 4P3 = 4 ! = 24 (ii) The first prize can be given away in 4 ways as it may be given to anyone of the 4 boys. The second prize can also be given away in 4 ways, since it may be obtained by the boy who has already received a prize. Similarly, third prize can be given away in 4 ways. Hence, the number of ways in which all the prizes can be given away = 4x4x4 =4 3 =64 85 (iii) Since any one of the 4 boys may get all the prizes. So, the number of ways in which a boy get all the 3 prizes = 4. So, the number of ways in which a boy does not get all the prizes = 64-4=60 3.2.5 Circular Permutations: We have seen that the number of permutations of n different things taken all together is n\, where each permutation is a different arrangement of the n things in a row, or a straight line. These permutations are called linear permutations or simply permutations. A circular permutation is one in which the things are arranged along a circle. It is also called closed permutation. Theorem 3.4: The number of circular permutations of n distinct objects is (n- 1)! Proof: Letai, Q2, •■• , cin-h a n be n distinct objects. Let the total number of circular permutations be x. Consider one of these x permutations as shown in figure. Clearly this circular permutation provides n near permutations as given below a\, ci2, ai, ... , a n - \, a n Cl2, (23, «4, «3, «4, a$, a n ,a\, C12, a n > a 2 an -I Fig. 3. 1 Thus, each circular permutation gives n linear permutations. But there are x circular permutations. So, total number of linear permutations is xn. But the number of linear permutations of n distinct objects is n\. :. xn = n\ x = nl_ n x = (n - 1) ! .". The total number of circular permutations of n distinct objects is (n - 1)! Note: In the above theorem anti-clockwise and clockwise order of arrangements are considered as distinct permutations. 86 Difference between clockwise and anti-clockwise arrangements: Consider the following circular permutations: Fig. 3. 2 Fig. 3. 3 We observe that in both, the order of the circular arrangement is a u a 2 , a 3 , a 4 . In fig (3.2) the order is anti-clockwise, whereas in fig. (3.3) the order is clockwise. Thus the number of circular permutation of n things in which clockwise and anti-clockwise arrangements give rise to different permutations is (n - 1)! If there are n things and if the direction is not taken into consideration, the number of circular permutations is ^ (/i — 1)! Example 3.30: In how many ways 10 persons may be arranged in a (i) line (ii) circle? Solution: (i) The number of ways in which 10 persons can be arranged in a line = loPlO = 10! (ii) The number of ways in which 10 persons can be arranged in a circle = (10-1)!= 9! Example 3.31: In how many ways can 7 identical beads be stung into a ring? Solution: Since the arrangement is circular either clockwise arrangement or anti-clockwise arrangement may be considered. 1 6! .". The required number of ways = ~ (7- 1)! ="9" = 360 Example 3.32: In how many ways can 5 gentlemen and 5 ladies sit together at a round table, so that no two ladies may be together? Solution: The number of ways in which 5 gentlemen may be arranged is (5 - 1)! = 4! 87 Then the ladies may be arranged among themselves in 5! ways. Thus the total number of ways =4!x5!=24xl20 = 2880 Example 3.33: Find the number of ways in which 8 different flowers can be strung to form a garland so that 4 particular flowers are never separated. Solution: Considering 4 particular flowers as one flower, we have five flowers, which can be strung to form a garland in 4 ! ways. But 4 particular flowers can be arranged in 4! ways. .". Required number of ways = 4 ! x 4 ! = 576 EXERCISE 3.3 1 . Evaluate the following : (i) 5 P 3 (ii) 15P3 (iii) 5P5 (iv)25P20 (v) 9P5 2. If W P 4 = 20 . W P3 , findw. 3. If ioP r =5040, find the value of r. 4. If 56P(r + 6) :54P(r + 3) = 30800 : 1, find r 5. If P m stands for m P m , then prove that 1 + 1 .Pi + 2.P2 + 3.P3 + ... + n.P n = («+ 1)! 6. Prove that n V r = (n _ i)P r + r . („ _ i)P( r _ i). 7. Three men have 4 coats, 5 waistcoats and 6 caps. In how many ways can they wear them? 8. How many 4-letter words, with or without meaning, can be formed, out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed? 9. How many 3-digit numbers are there, with distinct digits, with each digit odd? 10. Find the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time. 11. How many different words can be formed with the letters of the word 'MISSISSIPPI'? 12. (i) How many different words can be formed with letters of the word 'HARYANA'? (ii) How many of these begin with H and end with N? 13. How many 4-digit numbers are there, when a digit may be repeated any number of times? 14. In how many ways 5 rings of different types can be worn in 4 fingers? 15. In how many ways can 8 students are seated in a (i) line (ii) circle? 16. In how many ways can a garland of 20 similar flowers are made? 3.3 Combinations: The word combination means selection. Suppose we are asked to make a selection of any two things from three things a, b and c, the different selections are ab, be, ac. Here there is no reference to the order in which they are selected. i.e. ab and ba denote the same selection. These selections are called combinations. Definition: A selection of any r things out of n things is called a combination of n things r at a time. Notation: The number of all combinations of n objects, taken r at a time is generally denoted by n C r or C(n,r) or (1 . We use the symbol n C r throughout our discussion. (Number of ways of selecting Thus n C r = j r ob j ects from „ bj e cts Difference between Permutation and Combination: 1. In a combination only selection is made whereas in a permutation not only a selection is made but also an arrangement in a definite order is considered. i.e. in a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. 2. Usually the number of permutation exceeds the number of combinations. 3. Each combination corresponds to many permutations. Combinations of n different things taken r at a time: Theorem 3.5: The number of all combinations of n distinct objects, taken r at a time is n\ given by n C r = ~ — — r b J n (n-r) ! r! Proof: Let the number of combinations of n distinct objects, taken r at a time be denoted by n C r . Each of these combinations contains r things and all these things are permuted among themselves. .". The number of permutations obtained is r ! Hence from all the n C r combinations we get n C r x r! permutations. But this gives all the permutations of n things taken r at a time i.e. n P r . Hence, n C r . r ! = „P r ■ • nW — «P. r r "'■ < r, M ' i.Pr = (n-r)\r\ \' n r (n-r)\ Properties (1)„C„=1 (2)„Co=l (3) n C r = n C n - r 0<r<n Proof: n\ (1) We know that n C r = (n - r)\ r\ Putting r = n, we have „C W = (n _ n)!n! = q!^! = 1 (2) Putting r = 0, we have n\ nj (n-0) !0! = «! m! m! "^O - /„_mim - „! - 1 (3) We have „C W _ r = N - ... / \ (« - r)! r! (n — r)\ \n— n — r )\ = rr-r Note: The above property can be restated as follows : If x and y are non-negative integers such that x + y = n, then n C x = n Cy (4) If n and r are positive integers such that r < n, then n C r + n C( r - l) = («+l)C r Proof: We have rr~-r + rp~-(r— 1) — (n-r)!r! („ _ —{), (r _ iy {n-r)\r\ (n-r+l) !(r-l) + (n-r) !r{(r-l)!} (n - r + 1) {(n - r)\ (r- 1)! 90 (1 1 (w-r)!(r-l)! [r n-r+1 n\ In — r + 1 + r (n-r)\(r-l)\ [r(n-r+l) n\ \ n + 1 (n-r)\{r-\)\ [r(n - r + I) (n+l) \n\] ~ (n - r + 1) (n - r)\ r(r - 1)! (m+1)! ~~ (n -r+ 1)! r! (ra+1)! ~~ (« + 1 - r) ! r! = in + l)C r (5) If n and r are positive integers such that 1 < r < n, then n C r = ~ (n-l)C(r-l) Proof: C - " ! npi-1)! [(n-l)-(r-l)]!r(r-l)l n (»-!)! r [(„-l)-(r-l)]!(r-l)] r (n-l)C(r-l) (6)If 1 < r < n, then « . („ _ i)C( r - 1) = (n - r + 1) . „C( r _ i) Proof: We have n. (n . 1)C(r _ 1} = „ {^ _ - ^^ - _ - 77 (n-r)!(r-l)! (n- r + \)n\ (n-r+l)(n-r)!(r-l)! 91 ;/ (n-r+l)\ (r-1)!. = (n-r+Y) = ( "- r+1) >-r-l)!(r-l)! = (n-r+ 1) . w C( r _ i) (7) For any positive integers x and y, n C x = n C y ^ x = y or x + y = n Proof: We have n C x = n C y — > nS-x = rp-y = nS-'in - y) V '■' rr-y = n^-(n - y)\ =^>x = y or x = n-y =>x = y or x + y = n Note: If n C x = nCy and x ^ y, then x + y = n Example 3.34: Evaluate the following : 5 (ii) I 5C r (i) 6C 3 Solution: 0) (ii) r=\ 6C3 = 6P3 6x5x4 3! 1x2x3 = 20 Z 5C r = 5C1 + 5C2 + 5C3 + 5C4 + 5C5 r=l = 5 + 10+10 + 5+1=31 Example 3.35: If W C4 = n C(, , find i2C n Solution: W C4 = n C(, => « = 4 + 6 = 10 Now i 2 C„ = 12C10 12x11 = 12C(12 - 10) = 12C2 = i x 2 = 66 Example 3.36: If 15CV : isC( r _ i) = 11:5, find r 92 Solution: „ - 15Q 11 r-lj-xx., - 15 C (r _i) - 15! 5 r!(15-r)! 11 15! 5 (r-l)!(15-r+ 1)! 15! (r-l)!(16-r)! 11 r!(15-r)! x 15! (r-l)!(16-r){(15-r)!} 5 11 r(r -1)! (15 - r)! 16 -r r 5 11 5 => 5(16 -r) = llr => 80 = 16r => r= 5 Example 3.37:Show that the product of r consecutive integers is divisible by r! Solution: Let the r consecutive integers be « + l,n + 2,n + 3, ...,n + r Hence their product = (n+ 1) (n + 2) (n + 3) ... (n + r) _ 1.2.3. .. n.(n+ l)(w + 2) ... (w + r) 1.2.3... n _ (n + r) ! = re! their product _ (n + r) ! r! ~ n\r\ = (n + r )C r which is an integer. .". The product of r consecutive integers is divisible by r! Example 3.38: Let r and n be positive integers such that 1 < r < n. Then prove the following : «C r n — r + i 77 C(r-l) r «C r r!(n-r)! Solution: n C(r-l) (r-l)!(n-r+l)! 93 n\ (r-l)!(w-r+l)! ~ r\(n-r)\ * n\ (r-l)!Qi-r+l) {(n-r)!} r(r-l)!(w-r)! n — r + 1 r Example 3.39 : If n P r = n P( r + i) and n C r = w C( r - 1) , find the values of n and r Solution: n! (1) w P r - n P(r+ 1) => (n-r)! 1 ~ fn-r-1)! 1 => (« -r)(w-r-l)! ~ (n-r-l)\ => rr-r — n*-(r — 1) — ^ « - r n! = 1 ... i r! (n - r) ! ~ (r-l)\(n-r+l)\ «! n\ ^> r(r-l)!(w-r)! ~ (r-l)!(n-r+l) {(«- -r)!} 1 1 r ~ n-r+l => n - r + 1 = r => w-2r =-1 ...(2) Solving (1) and (2) we get n = 3 and r = 2 EXERCISE 3.4 1. Evaluate the following: (i) 10C8 (ii) 100C98 (iii) 75C75 2. If n Cio = w Ci2, find23C n 3. If sCr-7C3 = 7C2, find r 4. If i 6 C 4 =i 6 C r + 2,find r C2 20 5. Find n if (1) 2 . n C?, = -y „C2 (11) „C(„ _ 4) =70 6. If („ + 2)Cs : (n - 2)?4 = 57 : 16, find n. 7. If 28C2/- : 24C(2r - 4) = 225 : 11, find r. 94 Practical problems on Combinations Example 3.40: From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways this can be done? Solution: There are 15 players in a group. We have to select 11 players from the group. .". The required number of ways = 15C1 1 15xl4xl3x 12 ,„„ 15Cl1 = 1x2x3x4 = 1365wa y s Example 3.41: How many different teams of 8, consisting of 5 boys and 3 girls can be made from 25 boys and 10 girls? Solution: 5 boys out of 25 boys can be selected in 25C5 ways. 3 girls out of 10 girls can be selected in 10C3 ways. .". The required number of teams = 25C5 x 10C3 = 6375600 Example 3.42: How many triangles can be formed by joining the vertices of a hexagon? Solution: There are 6 vertices of a hexagon. One triangle is formed by selecting a group of 3 vertices from given 6 vertices. This can be done in 6C3 ways. 6! .•. Number of triangles = 6C3 = 3T37 = 20 Example 3.43: A class contains 12 boys and 10 girls. From the class 10 students are to be chosen for a competition under the condition that atleast 4 boys and atleast 4 girls must be represented. The 2 girls who won the prizes last year should be included. In how many ways can the selection are made? Solution: There are 12 boys and 10 girls. From these we have to select 10 students. Since two girls who won the prizes last year are to be included in every selection. 95 So, we have to select 8 students from 12 boys and 8 girls, choosing atleast 4 boys and atleast 2 girls. The selection can be formed by choosing (i) 6 boys and 2 girls (ii) 5 boys and 3 girls (iii) 4 boys and 4 girls .•. Required number of ways = (12C6 x 8^2) + (12C5 x 8C3) + (12C4 x 8^4) = (924 x 28) + (792 x 56) + (495 x 70) = 25872 + 44352 + 34650 = 104874 Example 3.44: How many diagonals are there in a polygon? Solution: A polygon of n sides has n vertices. By joining any two vertices of a polygon, we obtain either a side or a diagonal of the polygon. Number of line segments obtained by] joining the vertices of a n sided f=Number of ways of selecting 2 out of n polygon taken two at a time J _ n(n - 1) — nS-2 — 2 Out of these lines, n lines are the sides of the polygon. n(n — 1) .•. Number of diagonals of the polygon = ~ - n n(n - 3) 2 Example 3.45 How many different sections of 4 books can be made from 10 different books, if (i) there is no restriction (ii) two particular books are always selected; (iii) two particular books are never selected? Solution: 10! (i) The total number of ways of selecting 4 books out of 10 = ioC4= 4 , fi , = 210 (ii) If two particular books are always selected. This means two books are selected out of the remaining 8 books 8! .". Required number of ways = 8C2 = tTaT = 28 (iii) If two particular books are never selected This means four books are selected out of the remaining 8 books. 8! .". Required number of ways = 8C4 = ~T\ 41 = 70 96 Example 3.46: In how many ways players for a cricket team can be selected from a group of 25 players containing 10 batsmen, 8 bowlers, 5 all-rounders and 2 wicket keepers? Assume that the team requires 5 batsmen, 3 all-rounder, 2 bowlers and 1 wicket keeper. Solution: The selection of team is divided into 4 phases: (i) Selection of 5 batsmen out of 10. This can be done in 10C5 ways, (ii) Selection of 3 all-rounders out of 5. This can be done in 5C3 ways. (iii)Selection of 2 bowlers out of 8. This can be done in 8C2 ways. (iv)Selection of one wicket keeper out of 2. This can be done in 2C1 ways. .". The team can be selected in 10C5 x 5C3 x gC2 x 2C4 ways = 252 x 10 x 28 x 2 ways = 141120 ways Example 3.47: Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines (ii) triangles can be formed by joining them? Solution: (i) Number of straight lines formed joining the 18 points, taking 2 at a time = I8C2 =153 Number of straight lines formed by joining the 5 points, taking 2 at a time = 5C2 =10 But 5 collinear points, when joined pairwise give only one line. .". Required number of straight lines = 153 - 10 + 1 = 144 (ii) Number of triangles formed by joining the 18 points, taken 3 at a time = 18C3 = 816 Number of triangles formed by joining the 5 points, taken 3 at a time = 5C3 = 10 But 5 collinear points cannot form a triangle when taken 3 at a time. .". Required number of triangles = 816-10 = 806 EXERCISE 3.5 1. If there are 12 persons in a party, and if each two of them shake hands with each other, how many handshakes happen in the party? 97 2. In how many ways a committee of 5 members can be selected from 6 men and 5 women, consisting of 3 men and 2 women? 3. How many triangles can be obtained by joining 12 points, five of, which are collinear? 4. A box contains 5 different red and 6 different white balls. In how many ways 6 balls be selected so that there are atleast two balls of each colour? 5. In how many ways can a cricket team of eleven be chosen out of a batch of 15 players if (i) there is no restriction on the selection (ii) a particular player is always chosen; (iii) a particular player is never chosen? 6. A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions. 7. There are 10 points in a plane, no three of which are in the same straight line, excepting 4 points, which are collinear. Find the (i) the number of straight lines obtained from the pairs of these points (ii) number of triangles that can be formed with the vertices as these points. 8. In how many ways can 21 identical books on Tamil and 19 identical books on English be placed in a row on a shelf so that two books on English may not be together? 9. From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can they be chosen? 3.4 Mathematical Induction: Introduction: The name 'Mathematical induction' in the sense in which we have given here, was first used by the English Mathematician Augustus De-Morgan (1809 - 1871) in his article on 'Induction Mathematics' in 1938. However the originator of the Principle of Induction was Italian Mathematician Francesco Mau Rolycus (1494 - 1575). The Indian Mathematician Bhaskara (1153 A.D) had also used traces of 'Mathematical Induction' in his writings. "Induction is the process of inferring a general statement from the truth of particular cases". 98 For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 7 + 3 and so on. From these cases one may make a general statement "every even integer except 2 can be expressed as a sum of two prime numbers. There are hundreds of particular cases where this is known to be true. But we cannot conclude that this statement is true unless it is proved. Such a statement inferred from particular cases is called a conjecture. A conjecture remains a conjecture until it is proved or disproved. Let the conjecture be a statement involving natural numbers. Then a method to prove a general statement after it is known to be true in some particular cases is the principle of mathematical induction. Mathematical induction is a principle by which one can conclude that a statement is true for all positive integers, after proving certain related propositions. The Principle of Mathematical Induction: Corresponding to each positive integer n let there be a statement or proposition P(n). If (i) P(l)istrue, and (ii) P(k + 1) is true whenever P(k) is true, then P(n) is true for all positive integers n. We shall not prove this principle here, but we shall illustrate it by some examples. Working rules for using principle of mathematical induction: Step (1) : Show that the result is true for n = 1. Step (2) : Assume the validity of the result for n equal to some arbitrary but fixed natural number, say k. Step (3) : Show that the result is also true for n = k + 1 . Step (4) : Conclude that the result holds for all natural numbers. 2 Example 3.48: Prove by mathematical induction n + n is even. 2 Solution: Let P(w) denote the statement "n + n is even" Step (1): Put n = 1 2 2 n + n = 1+1 = 2, which is even .'. P(l) is true Step (2): Let us assume that the statement be true for n = k 99 (i.e.) assume P(k) be true. 2 (i.e.) assume "k + k is even" be true ... (1) Step (3): To prove P(k + 1) is true. 2 (i.e.) to prove (k + 1) + (k + 1) is even Consider (k + l) 2 + (k + 1) = £ 2 + 2k + 1 + fc + 1 = k 2 + 2k + k + 2 = (k 2 + k) + 2(k + 1) = an even number + 2(k + 1), from (1) = sum of two even numbers = an even number .-. P(k + 1) is true. Thus if P(k) is true, then P(k + 1) is also true. Step (4): .". By the principle of Mathematical induction, P(w) is true for all weN. 2 i.e. n + n is even for all weN. n(n + 1) Example 3.49: Prove by Mathematical induction l+2 + 3 + ...+ « = j > weN nin + 1) Solution: Let P(w) denote the statement : "l+2 + 3 + ... + « = ~ " Put n = 1 P(l) is the statement : 1 = ~ 1(2) 2 1 = 1 1 = .'. P(l) is true Now assume that the statement be true for n = k. (i.e.) assume P(k) be true. k(k+ 1) (i.e.) assume I + 2 + 3 + ... + k = — ~ — • • • (1) b e true To prove P(fc + 1) is true (i.e.) to provel +2 + 3 + ... + k + (k+ 1)= ~ i s true ' k(k + 1) [1+2 + 3 + ... + k] + (k+l) = 2 ' + (k+l) from(l) 100 k(k + 1) + 2(k + 1) 2 (k+l)(k + 2) 2 :. P(k + 1) is true. Thus if P(k) is true, then P(k + 1) is true. By the principle of Mathematical induction, P(n) is true for all neN n(n + 1) .-. 1+2 + 3 + ... + /!= 2 forallneN Example 3.50: Prove by Mathematical induction ,2 -2 .2^ 2 w(w + 1) (2w + 1) 1 +2 +3 +...+n = t forallneN Solution: Let P(n) denote the statement "1 +2 +3 +...+« = 7 " Put n = 1 2 1(1 + 1) [2(1) +1] P(l) is the statement : 1 = 7 . K2) (3) 1 - 6 1 = 1 .". P(l) is true. Now assume that the statement be true for n = k. (i.e.) assume P(k) be true. ... ,2^,2 .2^ .2 k(k+l)(2k+l) (i.e.) 1 +2 +3 +...+& = t ■•■(!) To prove : P(k + 1) is true ,. . ,2 2 2 ,2 ., ,,2 (k+l)(k + 2)(2k + 3) . (i.e.) to prove: 1 +2 +3 +...+K +(fe+l) = t is true. [l 2 + 2 2 + 3 2 + ... +fc 2 ] + (^l) 2 = ^ +1) 6 (2/c+1) + (^ + l) 2 k(k + I) (2k +1) + 6(k + I) 2 6 _ (k + I) [k(2k + I) + 6(k + I)] 6 (yt + 1) (2/t 2 + 7/t + 6) 101 i 2 ^9 2 ^ 2 ^ a 2 .^n 2 (k+l)(k + 2)(2k + 3) 1 +2 +3 +...+K +(k+1) = 7 .-. P(& + 1) is true Thus if P(k) is true, then P(k + 1) is true. By the principle of Mathematical induction, P(n) is true for all neN r ^ ! 2 ^o 2 ^ ^ 2 "(w+l)(2w+l) r „ M (i.e.) 1 + 2 + ...+« = t for all neN Example 3.51: Prove by Mathematical induction l2 + 23 + 3A + ... + n(n + l) = n(n+1) 3 (n + 2 \ neN. Solution: n(n + 1) (n + 2) Let P(n) denote the statement "1.2+2.3 + 3.4 +. . .+ n(n + 1)= — y " Put « = 1 P(l) is the statement : 1(1 + 1) = 3 K2) = ^P ,2(3) z_ 3 2 = 2 .". P(l) is true. Now assume that the statement be true for n = k. (i.e.) assume P(k) be true k(k + 1) (k + 2) (i.e.) assume 1.2 + 2.3 + 3.4+...+ k(k+ 1) = — 3 L be true To prove : P(k + 1) is true i.e. to prove : 1.2 + 2.3 + 3.4+...,+ ^+l) + (^+l)(^ + 2) = (/c+1)(/C 3 2)(/c + 3) Consider 1.2 + 2.3 + 3.4 + . . .,+ k(k + 1) + (k + 1) (k + 2) = [l.2 + 2.3 + ...+fc(fc+l)] + (fc+l)(fc + 2) fc(fc+lHfc+2) „ 1W , „ N = — y + (k + 1) (k + 2) fc(fc+l)(fc + 2) + 3(fc+l)(fc + 2) 3 (&+!)(* + 2) (ft + 3) 3 102 .-. P(k + 1) is true Thus if P(k) is true, P(k + 1) is true. By the principle of Mathematical induction, P(n) is true for all neN. , „ „ -, „ „ , ^ re(n + 1) (n + 2) 1.2 + 2.3 + 3.4+ ... +n(n+ 1) =— y L 3« Example 3.52: Prove by Mathematical induction 2 - 1 is divisible by 7, for all natural numbers n. Solution: 3« Let P(n) denote the statement "2 - 1 is divisible by 7" Put n = 1 Then P(l) is the statement : 2 3(1) - 1 = 2 3 - 1 = 8-1 = 7, which is divisible by 7 .". P(l) is true Now assume that the statement be true for n = k 3k (i.e.) assume P(k) be true, (i.e.) "2 - 1 is divisible by 7" be true Now to prove P(fc + 1) is true, (i.e.) to prove 2 A - 1 is divisible by 7 Consider 2 Xk+l)_ -l=2 3fe+3 -l = 2 3k . 2 3 - 1 = 2 3fc . 8 - 1 3£ = 2 .8-1+8-8 (add and subtract 8) 3k = (2 - 1) 8 + 8 - 1 3k = (2 - 1) 8 + 7 = a multiple of 7 + 7 = a multiple of 7 :. 2 - 1 is divisible by 7 .-. P(& + 1) is true Thus if P(k) is true, then P(k + 1) is true. By the principle of Mathematical induction, P(n) is true for all we N 3« :. 2 - 1 is divisible by 7 for all natural numbers n. n n Example 3.53: Prove by Mathematical induction that a - b is divisible by (a-b) for all n e N n n Solution: Let P(w) denote the statement "a - b is divisible by a - b". 103 Put n = 1 Then P(l) is the statement : a -b = a - b is divisible by a - b :. P(l) is true. Now assume that the statement be true for n = k. k k (i.e.) assume P(k) be true, (i.e.) a - b is divisible by (a - b) be true. a - b => — = c (say) where ceN a - b J => a - b = c(a - b) k k => a = b +c(a-b) ... (1) k + I k+ 1 Now to prove P(k + 1) is true, (i.e.) to prove : a - b is divisible by a - b Consider a - b = a . a- b b = [b k + c(a - b)] a-b b k k = b a + ac(a - b) - b b k = b (a — b) + ac (a — b) k = (a-b)(b + ac) is divisible by (a - b) :. P(k + 1) is true. By the principle of Mathematical induction, P(w) is true for all we N n n :. a - b is divisible by a - b for all n e N EXERCISE 3.6 Prove the following by the principle of Mathematical Induction. (1) (In + 1) (7m - 1) is an odd number for all tie N (2) 2 + 4 + 6 + 8 + . . . + In = n (n + 1) (3) l + 3 + 5+...+(2«-l) = « 2 (4) 1+4 + 7+ ...+(3n-2)= " 2 ~ - (5) 4 + 8+12+...+4w = 2w(«+l) ,~ ,3 .3 ,3 3 n 2 (n + l) 2 (6) 1 +2 +3 + ... +n = ^ ,_. I J_ J_ J_ . J_ (7) 2 + 9 2 + 7? + ■■■ + i n ~ 2" 104 (8) In the arithmetic progression a, a + d, a + 2d, . . . i th . . . . , then term is a + (n - \)d 2n (9) 5 - 1 is divisible by 24 for all mg N (10) 10 2 " ~ l + 1 is divisible by 1 1. (1 1) n(n + 1) (n + 2) is divisible by 6 where n is a natural number. 3 2 (12) ThesumS„ = « + 3« + 5n + 3 is divisible by 3 for all n e N (13) 7 2 " + 16« - 1 is divisible by 64 (14) 2" > w for all we N 3.5 Binomial Theorem: Introduction: A BINOMIAL is an algebraic expression of two terms which are connected by the operation '+' (or) '-' 3 For example, x + 2y, x - y, x +4y, a + b etc.. are binomials. Expansion of Binomials with positive Integral Index: We have already learnt how to multiply a binomial by itself. Finding squares and cubes of a binomial by actual multiplication is not difficult. But the process of finding the expansion of binomials with higher powers 10 17 25 such as (x + a) , (x + a) ,(x+ a) etc becomes more difficult. Therefore we look for a general formula which will help us in finding the expansion of binomials with higher powers. We know that (x + a) =x + a = jCrj x a + jCi x a ,2 2 . 2 r 20 . 11 „ 02 (x + a) =x +2ax + a = 2^-0% a + iy.\x a + 2^-2* a 33 2 23 30 21 12 03 (x + a) =x + 3x a + 3xa +a = 3C0* a + j,C\x a + 3C2* a + 3C3X a 44322 34 40 31 22 13 04 (x+a) =x +4x +6x a +4xa +a =<\C<yc a +4C1X a +4C2X a +4C3X a +4C4X a n For n = 1, 2, 3, 4 the expansion of (x + a) has been expressed in a very systematic manner in terms of combinatorial coefficients. The above n expressions suggest the conjecture that (x + a) should be expressible in the form, 105 , ,« „ « _ n — 1 1 „ 1 « - 1 _ « (x + a) = nto i a + nL\x a + ...+nC M _ix a + nLnxa In fact, this conjecture is proved to be true and we establish it by using the principle of mathematical induction. Theorem 3.6: (Binomial theorem for a Positive Integral Index) Statement: For any natural number n . .n „ n „ n - 1 1 „ n -r r , (x + a) = wCo x a + nC ix a + . . . + nLr x a + ... „ ln-1 « + wC M _ \x a + nLn x a Proof: We shall prove the theorem by the principle of mathematical induction. Let P(w) denote the statement : . .« _ « _ n-1 1 ^ rc-r r (x + a) = uLqx a + mC]x a +... + nLr x a + ... „ 1 n - 1 „ « + «C M _ i x a + nLn x a Step (1) : Put n = 1 Then P(l) is the statement : (x + a) = iQ>x a + 1C1 x a x + a = x + a .'. P(l) is true Step (2): Now assume that the statement be true for n = k (i.e.) assume P(k) be true. , .k ,„ k , „ fe-1 1 ,_, fc-2 2 ,_, k-r r _, Ofe (x + a) =/c(_ox a +/cC]x a +/cL2X a +...+ kL r x a +...+ £(_£x a be true ... (1) Step (3): Now to prove P(k + 1) is true (i.e.) To prove: .K+l _ fc+1 _ (jfc+l)-ll _ (*+l)-2 2 (x + a) = (jfc+l)Co* +(Jt+i)Cix a+(jt+i)C2x a +... _ (fc+ l)-r r _ fe+1 + (fc+l)C,x a +... + (jfc+i)C(jfc+i)a fe+ 1 A: Consider (x + a) = (x + a) (x + a) rI _ jfe ,„ i-11 ,„ k -2 2 ._ fe-(r-l) (r-1) = [kCqx +/cCix a +/cC2X a + ... + kC( r _i)X a + kC r x a + ... + kC^a ] (x + a) 106 r .„ k+\ k \ k-12 k-r+2 r-1 = [kL.QX + kL\x a + W_2X a + ... + kL. r -\x a + kC r x a + ... + kCfcxa ] r ,„ k , „ k- 1 2 ,_ t-2 3 ,„ k-r+lr + [kLqx a + kL\x a +kL2X a + ...+ kL r _ix a ,_ k - r r+\ ,^ fc+ 1, + kL. r x a + . . . + kL-kCi J k + 1 k+ 1 fe fe-12 (x + a) = fcCox + (kC{ + kCo) x .a + (kC2 + kC[) x a + ... + (kC r + kC r - \)x a+...+kCka •••(2) We know that kC r + kC r - \ = (k+ 1)0- Put r= 1,2,3, ... etc. kC\ +kCo = (fc+i)Ci &C2 + fcCi = (fc+i)C2 fcC r + &C r -i = ( t + i)C r for 1 < r < k kCo = 1 = (fc+i)Co fcCfe = 1 = (jfc+i)C(fc+l) .". (2) becomes . . fc + 1 ^ fc + 1 ^fc „ 1-12 (x + a) = (jt+1) Cox + (fe+ i)Cix a + (fe+i)C2X a _ fc+ 1 - r r _ yt+1 + •■• +(k+l)C r x a + ... + (Jt+i)C(jt+i)a .-. P(£ + 1) is true Thus if P(&) is true, P(& + 1) is true. .". By the principle of mathematical induction P(re) is true for all neN . .« _ « _ n — 1 1 ^ n- r r (x + a) = uLqx a + nL\x a +... + reCrx a + ... _, 1 re - 1 _, n , „ .. + reC M _ i x a + reCre x a tor all n e N Some observations: 1. In the expansion . .« _ « _ re — 1 1 ^ n-r r (x + a) = uLqx a + nL\x a +... + reCrx a + ... _ 1 re — 1 _ « . , • ^ n- r r + reC M _ i x a + reCre x a , the general term is nL r x a . Since this is nothing but the (r + 1) term, it is denoted by T r + i r + i = nL r x a . 2. The (re + 1) termisT n+ i = nC n x n ~ n a n = nC n a n , the last term. Thus there are (n + 1) terms in the expansion of (x + a) n 107 3. The degree of x in each term decreases while that of "a" increases such that the sum of the powers in each term is equal to n. n We can write (x + a) = £ nC r x a r=0 4. hCq, «Ci, nC2, • ••, nC r , ... , nC n axe called binomial coefficients. They are also written as Co, Ci , C2, ■ • • , C n . 5. From the relation nC r = nC n _ r , we see that the coefficients of terms equidistant from the beginning and the end are equal. 6. The binomial coefficients of the various terms of the expansion of n (x + a) for n = 1, 2, 3, . . . form a pattern. Binomials Binomial coefficients (x + a) l (x + a) ll (x + a) 2 l 2 l (x + a) 3 l 3 3 l (x + a) 4 l 4 6 4 l (x + a) 5 l 5 10 10 5 l This arrangement of the binomial coefficients is known as Pascal's triangle after the French mathematician Blaise Pascal (1623 - 1662). The numbers in any row can be obtained by the following rule. The first and last numbers are l each. The other numbers are obtained by adding the left and right numbers in the previous row. I, 1 + 4 = 5, 4 + 6 = 10, 6 + 4=10, 4+1=5, 1 Some Particular Expansions: In the expansion . .« _ n _ n -1 1 _ n -r r (x + a) =uLqx a + nL\x a +...+ nLrx a +... „ 1 n - 1 „ n .,, + riL n - 1 x a + nLn x a ... (1) 1. If we put - a in the place of a we get n n n-lln-2 2 :.(x — a) = nLox -nL\x a + nL2x a -... , 1x r _, n - r r , ,,« „ n + (-1) nL r x a +... + (— 1) nL n a 108 We note that the signs of the terms are positive and negative alternatively. 2. If we put 1 in the place of a in (1) we get, Yl 2 Y Yl (1 + x) = 1 + nC\x + nC2X +...+nC r x +...+nC n x •••(2) 3. If we put - x in the place of x in (2) we get Yl 2 X Y Yl Yl (1 — x) =l-nC\x + nC2X -... + (-1) nC r x +...+(-1) nC n x Middle Term: Yl The number of terms in the expansion of (x + a) depends upon the index n. The index is either even (or) odd. Let us find the middle terms. Case (i) : n is even The number of terms in the expansion is (n + 1), which is odd. Therefore, there is only one middle term and it is given by T« 2 +1 Case (ii) : n is odd The number of terms in the expansion is (n + 1), which is even. Therefore, there are two middle terms and they are given by T« + 1 and 2 Tn + 3 2 Particular Terms: Sometimes a particular term satisfying certain conditions is required in Yl Yl the binomial expansion of (x + a) . This can be done by expanding (x + a) and then locating the required term. Generally this becomes a tedious task, when the index n is large. In such cases, we begin by evaluating the general term T r+ i and then finding the values of r by assuming T r+ i to be the required term. To get the term independent of x, we put the power of x equal to zero and get the value of r for which the term is independent of x. Putting this value of r in T r+ i, we get the term independent of x. 5 ( 3^ 4 Example 3.54:Find the expansion of : (i) (2x + 3y) (ii) ( 2x — — \ Solution: (i) (2x + 3y) 5 = 5 C (2x) 5 (3yf + 5C1 (2x) 4 (3y) 1 + 5 C 2 (2x) 3 (3y) 2 + 5 C 3 (2x) 2 (3y) 3 + 5C4 (2x) 1 (3y) 4 + 5C5 (2x)° (3y) 5 = l(32)x 5 (1) + 5(16x 4 ) (3;y) + 10(8x 3 ) (9y 2 ) 109 + 10(4jc 2 ) (21y 3 ) + 5(2x) (8l/) + (1) (1) (243/) = 32jc 5 + 240x 4 y + 720x 3 y 2 + 1080xV + 810xy 4 + 243y 5 (Hi \2x 2 -^] = 4 C (2x 2 ) 4 (-£) + 4 Ci(2x 2 ) 3 (-|) 2 N 2 ( 3^ 2 „ _ 2 N 1 T 3V ^ ,~1S>( 3^ 4 + 4 C 2 (2x) I- -J + 4 C 3 (2x) {--) + 4 C 4 (2x) v v = (1) 16x 8 (l) + 4(8x 6 ) (- 1) + 6(4/) (J) + 4(2* 2 ) [- |j + (D(1)^ = 16jc 8 - 96x 5 + 216jc 2 - — + ~j x Example 3.55: Using binomial theorem, find the 7 power of 1 1 . Solution: ll 7 = (1 + 10) 7 = 7C0 (l) 7 (10)°+ 7 Ci (l) 6 (10) 1 + 7 C2(1) 5 (10) 2 +7C 3 (1) 4 (10) 3 +7C 4 (1) 3 (10) 4 + 7C5 (l) 2 (10) 5 + 7 C 6 (l) 1 (10) 6 + 7C7 (D° (10) 7 7x6 2 7x6x5 37x6x5 4 7x6 5 6 7 = 1+70+— rlO + . . - 10 + - . . 10 4 + --r 10 +7(10)° + 10 7 1x2 1x2x3 1x2x3 1x2 = 1 + 70 + 2100 + 35000 + 350000 + 2100000 + 7000000 + 10000000 = 19487171 r \ X1 Example 3.56: Find the coefficient of x in the expansion of \x + ~~y I x Solution: n 17 In the expansion of [ x + 3 , the general term is x ) T r+ i = nC r x r -j I r 17 -4r Let T r + 1 be the term containing x then, 17-4r=5 =^> r = 3 110 •'• Tr+ 1 = T3+ i 17-4(3) 5 = 17C3 x K ' = 680x .". coefficient of x = 680 ( 2V Example 3.57: Find the constant term in the expansion of K/x - ~~2 \ x ) Solution: 10 In the expansion of [ yfx - ~j T r +i = 10Cr (V*) - r K x ) = 10 -r 10C r x 2 2r - 10Cr (- 2) x 10 -5r -2r = K)C r (-2) x 2 Let T r + 1 be the constant term Then, 10 -5r 2 => r=2 10-5(2) l C 2 (-2) x 2 .". The constant term - = 10x9 . . x 4 xx 1x2 - 18 f7 Example 3.58: If we N, in the expansion of (1 + x) prove the following : n (i) Sum of the binomial coefficients = 2 (ii) Sum of the coefficients of odd terms = Sum of the coefficients of even r,n- 1 terms = 2 Solution: The coefficients «Co, nC[, nQ% ... , nC n in the expansion of ft (1 + x) are called the binomial coefficients, we write them as Co, Ci, C2, • • • C n , n 2 (1+x) = C0 + C1X + C2X +...+■ It is an identity in x and so it is true for all values of x. II 2 T Yl (1+x) = C0 + C1X + C2X +...+C r x +...+C n x 111 Putting x = 1 we get 2 n = C + Ci+C 2 +...+C w ...(1) put x = - 1 o = Co-Ci + c 2 -c 3 + ...(-i) n c n =^Q) + C2 + C4+ ...= C1+C3 + C5+... It is enough to prove that C0 + C2 + C4+... = C1+C3 + C5+ ... =2 n ~ l Let C0 + C2 + C4+... = C1+C3 + C5+... =k ... (2) n From(l), C0 + C1+C2+ ... +C„ = 2 2k = 2 n From (2) k =2 n ~ l From (2), C0 + C2 + C4+... = Ci + C3 + C5 + ... = 2" ~ ! EXERCISE 3.7 (1) Expand the following by using binomial theorem (i) (3a + 5b) 5 (ii) (a - 2b) 5 (iii) (2x - 3x 2 ) 5 (iv) [x + -J (v) (x +2y ) (vi) (x -Jy + yyjx) (2) Evaluate the following: (i) (V2 + l) 5 + (V2 - l) 5 (ii) (V5 + l) 5 - ( V3 - l) 5 (iii) ( 1 + V5) 5 + (1 - ^/5) 5 (iv) ( 2sfa + 3) 6 + ( 2y[a - 3) 6 (v)(2 + V3) 7 -(2-a/3) 7 3 3 (3) Using Binomial theorem find the value of (101) and (99) . 3 (4) Using Binomial theorem find the value of (0.998) . (5) Find the middle term in the expansion of 2x 2 r fb xv 6 (i) \3x-^r (ii) 7 + 3 ; w u 16 r i\17 x m(-,-^i\ (iv)(.v-2y) B (v)f* + 4 *) 112 (6) Show that the middle term of 2 „ 1.3.5.7 ...(2« -1)2"/ (i)(l+x) is ~ { ....( if" . 1.3.5. ...(2W-1) (ii)(x + ^J is ~ { f if" . (-l) W .1.3.5.7....(2n-l) < m > {* ~ x) 1S n\ 2 11 (7) Find the coefficient of x in the expansion of I x- — J (8) Find the term independent of x (constant term) in the expansion of ( 2 if 2 (4x 2 3 f ( b- X1 (H 2 * + x) ^{—-Tx) W{ 9X -^ (9) In the expansion of (1 + x) , the coefficient of r and (r + l) 1 terms are in the ratio 1 : 6, find the value of r. If the coefficients are in A. P., find n. (10) If the coefficients of 5 ,6 and 7 terms in the expansion of (1 + x) 113 4. SEQUENCE AND SERIES 4.1 Introduction We hear statements such as "a sequence of events", "a series of tests before the board examination", "a cricket test match series". In all these statements the words "sequence" and "series" are used in the same sense. They are used to suggest a succession of things or events arranged in some order. In mathematics these words have special technical meanings. The word 'sequence' is used as in the common use of the term to convey the idea of a set of things in order, but the word "series" is used in a different sense. Let us consider the following example. A rabbit and a frog are jumping on the same direction. When they started they were one metre apart. The rabbit is jumping on the frog in order to catch it. At the same time the frog is jumping forward half of the earlier distance to avoid the catch. The jumping process is going on. Can the rabbit catch the frog? 1 IS «l I a, 1 R i * ' F, - R, j2_F,yR, • • • • ., » 1 R* 22 .F 2 ± r^f 4 Fig. 4. 1 Let a\, a% 03, a\ ... be the distances between the rabbit and the frog at the first, second, third, fourth instants etc,. The distance between the rabbit and the frog at the first instant is 1 metre. 1 1 J_ 1 J_ .-. a\ = 1 ; (22 = 2 ; a 3 = 4 = 2 ; a 4 = 3 = 3 Here a\, a% ai, ... form a sequence. There is a pattern behind the arrangement of a\, a2, 03 ... Now a n has the meaning, (i.e.) a n is the distance between the rabbit and the frog at the n instant When a n becomes the rabbit will catch the frog. i.e. the distance between the frog and the rabbit is zero when n —> go Further a n - 1 "2" 1 As n -> ■oc, a,, -> 114 At this stage the rabbit will catch the frog. This example suggests that for each natural number there is a unique real number, i.e. 1 2 3 ... n V* 4* V* V* a i ci2 03 ... a n 1111 1 = 1 2 2 1 4 2 2 2"" Consider the following list of numbers (a) 8, 15, 22, 29, (b) 6, 18, 54, 162, In the list (a) the first number is 8, the 2 n number is 15, the 3 r number is 22, and so on. Each number in the list is obtained by adding 7 to the previous number. In the list (b) the first number is 6, the 2 n number is 18, the 3 r number is 54 etc. Each number in the list is obtained by multiplying the previous number by 3. In these examples we observe the following: (i) A rule by which the elements are written (pattern). (ii) An ordering among the elements (order). Thus a sequence means an arrangement of numbers in a definite order according to some rule. 4.2 Sequence A sequence is a function from the set of natural numbers to the set of real numbers. If the sequence is denoted by the letter a, then the image of n e N under the sequence a is a(n) = a n . Since the domain for every sequence is the set of natural numbers, the images of 1, 2, 3, ... n .. . under the sequence a are denoted by a\, a2, a-^ ... a n , . . . respectively. Here a\, a2, aj, ...a n , ... form the sequence. "A sequence is represented by its range". Recursive formula A sequence may be described by specifying its first few terms and a formula to determine the other terms of the sequence in terms of its preceding terms. Such a formula is called as recursive formula. 115 For example, 1, 4, 5, 9, 14, . . ., is a sequence because each term (except the first two) is obtained by taking the sum of preceding two terms. The corresponding recursive formula is a n + 2 = a n + a n + l . n - 1 here «i=l, «2= 4 Terms of a sequence: The various numbers occurring in a sequence are called its terms. We denote the terms of a sequence by a\, a2, aj,, ... , a n , ... , the subscript denote the position of the term. The n term is called the general term of the sequence. For example, in the sequence 1, 3, 5, 7, ... 2« - 1, ... the 1 st term is 1, 2 n term is 3, and n l term is 2n - 1 Consider the following electrical circuit in which the resistors are indicated with saw-toothed lines. -"■Wv- 8 3 — ^•V'N- " Fig. 4. 2 If all the resistors in the circuit are 1 ohm with a current of 1 ampere then the voltage across the resistors are 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence there is no fixed pattern. But we can generate the terms of the sequence recursively using a relation. Every number after the second is obtained by the sum of the previous two terms, i.e. Vi = 1 V 2 = 1 v 4 = v 3 + v 2 v 5 = v 4 + v 3 V = V 'I? — v n + V„ 116 Thus the above sequence is given by the rule: Vi = l V2 = 1 Vn = Vn - 1 + Vn - 2 ; n > 3 This sequence is called Fibonacci sequence. The numbers occurring in this sequence are called Fibonacci numbers named after the Italian Mathematician Leonardo Fibonacci. Example 4.1: Find the 7 C term of the sequence whose n term is (- 1)" + Solution: ,n + l (n + 1 n + 1 n substituting n = 1, we get n7 + 1 Given a n = (- 1) n = 1 , we ai = (- l) 7 K j] - 7 4.3 Series For a finite sequence 1, 3, 5, 7, 9 the familiar operation of addition gives the symbol 1+3 + 5 + 7 + 9 which has the value 25. If we consider the infinite sequence 1, 3, 5, 7, ... then the symbol 1 + 3 + 5 + 7+.. . has no definite value, because when we add more and more terms the value steadily increases. 1 + 3 + 5 + 7 + 9+.. .is called an infinite series. Thus a series is obtained by adding the terms of a sequence. If a\, ai, aj,, ... a n ... is an infinite sequence then a\+ a2+ ... + a n + ... is 00 called an infinite series. It is also denoted by X a k k=\ If S„ = a\ +^2+ • • • + a n then S„ is called the n l partial sum of the series oo k=l 00 i Example 4.2 Find the n l partial sum of the series X ~ n=l 2 Solution: J_ J_ J_ S„- 2l + 22 +...+ 2 „ 117 11 1 1 and S„ + i =^j +^ +...+— +^TT $n+ 1 = S„ + „+ i Also we can write S„ + i as J_ J_ J_ 1 S« + l - i + 2 2 + • • • + 2 « + 2 n+[ 1 1 1 ...(1) S n+1 =2[l+S n ] ...(2) From (1) and (2) S„ + -—[ = 2^ + S «l Note: This can be obtained by using the idea of geometric series also. We know q(l-r ,T ) that the sum to n terms of a geometric series is S„ = " (1-r) Here a = 9 , n = n, r = ~z (< 1) ^n ~~ ^2 n« ■-§ 1 2" EXERCISE 4.1 (1) Write the first 5 terms of each of the following sequences: r\ t n«-U«+l ^ "(" + 5 ) (1) a n = (- 1) 5 (11) a n = 4 (iii) a n = - \\n + 10 1 /- 1 \« 2 1 - (- 1) M (v) «„ = 3 (vi) a n = — 118 (2) Find the indicated terms of the following sequences whose n term is 1 (n%\ (i)a n = 2+- ; a 5 , a 7 (n) a„ = cos I -y I ; a 4 , a 5 (in) an = „ ; «7 . «10 (iv)a„ = (-l) 2 , a 5 , a 8 (3) Find the first 6 terms of the sequence whose general term is 2 n — 1 if « is odd a„ = « 2 +l ., . it « is even 2 (4) Write the first five terms of the sequence given by (i) a\ = a2 = 2, a n = a n _ i - 1, n > 2 (ii) a\ = \, ci2 = 2, a n = a n _ i + a n _ 2, n > 2 (iii) a\ = 1 , a n = «a„ _ 1 , « > 2 (iv) ai = «2 = 1> a n = 2a« - 1 + 3a„ _ 2, « > 2 00 1 (5) Find the n partial sum of the series X ~ n=\ 3 00 (6) Find the sum of first n terms of the series X 5" n=\ CO 1 (7) Find the sum of 101 ! terms to 200 1 term of the series 2Z ~ n=\ 2 4.4 Some special types of sequences and their series (1) Arithmetic progression: An arithmetic progression (abbreviated as A.P) is a sequence of numbers in which each term, except the first, is obtained by adding a fixed number to the immediately preceding term. This fixed number is called the common difference, which is generally denoted by d. For example, 1, 3, 5, 7, ... is an A.P with common difference 2. (2) Arithmetic series: The series whose terms are in A.P is called an arithmetic series. For example, 1+3 + 5 + 7+.. . is an arithmetic series. 119 (3) Geometric progression A geometric progression (abbreviated as G.P.) is a sequence of numbers in which the first term is non-zero and each term, except the first is obtained by multiplying the term immediately preceeding it by a fixed non-zero number. This fixed number is called the common ratio and it is denoted by the letter V . The general form of a G.P. is a, ar, ar , ... , with a ^ and r * 0, the first term is 'a' (4) Geometric series: 2 n — \ The series a + ar + ar + ... + ar + ... is called a geometric series because the terms of the series are in G.P. Note that the geometric series is finite or infinite according as the corresponding G.P. consists of finite (or) infinite number of terms. (5) Harmonic progression: A sequence of non-zero numbers is said to be in harmonic progression (abbreviated as H.P.) if their reciprocals are in A. P. The general form of H.P is -, —^, —^ , ... , where a *0. n term of H.P. is T„ = : —. n a + (n- l)d For example the sequences 1, ■? , g , tt , ... is a H.P., since their reciprocals 1,5,9, 13, ... areinA.P. Note: There is no general formula for the sum to n terms of a H.P. as we have for A.P. and G.P. Example 4.3 If the 5 th and 12 th terms of a H.P. are 12 and 5 respectively, find the 15 th term. Solution: T = 1 n — " a + (n - \)d 1 1 Given T 5 =12^ a + (5 _ l)d =12 => ^^ = 12 a + Ad = J2 •■•(!) and Ti 2 = 5 => - - — _ j )d = 5 => —^ =5 => a+lld = j ...(2) 120 (2) - (1) (1) => ld = m a + 4 {eo =12 j_ 12 a + 60 => rf = 60 12 60 a = 60 ••• Tig = 1 1 a + (15 -l)rf J- +14x -L 60 + 14 x 60 J_ "15 60 Ti 5 = 4 60 15 4.5 Means of Progressions 4.5.1 Arithmetic mean A is called the arithmetic mean of the numbers a and b if and only if a, A, b are in A. P. If A is the A.M between a and b then a, A, b are in A.P => A-a = b - A => 2A = a + b A = - a + b A\, A2, •■• , A„ are called n arithmetic means between two given numbers a and & if and only if a, A\, A2, . . . A„, & are in A.P. Example 4.4 : Find the n arithmetic means between a and & and find their sum. Solution: Let A\, A2, • • • , A„ be the n A.Ms between a and b. Then by the definition of A.Ms a, Ai, A2, ... , A n , frarein A.P Let the common difference be cf. .'. Ai = a + cf, A2 = a + 2c/, A3 = a + 3d, ... , A n = a + nd and b = a + (n + l)d => (n + Y)d = b - a ■ a :d= n+\ .••A 1 =a + n+l b — a J2jb -a) A 2= fl+ n+ l »(£> - a) /ill — (A \ , 1 " H + 1 121 Sum of n A.Ms between a and b is b - a Ai+A 2 + ... +A n = a + n+l a + ' 2(b - a) n+l a +' n(b - a) n+l (b-a) = na + ~~r [I +2+ ... + ri] n + I L ' (b - a) n(n + 1) n(b - a) = na + 77TTT\ ■ t = na + 7. a + b (n+l) ■ 2 2na + nb - na na + nb 2 2 =n K. 2 Example 4.5: Prove that the sum of n arithmetic means between two numbers is n times the single A.M between them Solution: Let Aj, A2, ... , A„ be the n A.Ms between a and b. From the example (4.4) a + b Ai + A2 + A3 + . . . + A„ = n I 2 J = n x (A.M between a and b) = n (single A.M between a and b) Example 4.6: Insert four A.Ms between - 1 and 14. Solution: Let A1, A2, A3, A4 be the four A.Ms between - 1 and 14. By the definition - 1, Ai, A2, A3, A4, 14 are in A. P. Let d be the common difference. .-. A\= - 1 + d, ; A 2 = - 1 + 2d ; A3 = - 1 + 3d, ; A 4 = - 1 + Ad ; 14 = -l+5d :.d = 3 .-. Ai= - 1 + 3 = 2 ; A 2 = - 1+2 x 3 = 5 ; A3 = -1+3x3 = 8 ; A 4 = - 1 + 12 = 1 1 .-. The four A.Ms are 2, 5, 8 and 1 1. 4.5.2 Geometric Mean G is called the geometric mean of the numbers a and b if and only if a, G, b are in GP. G b => «=G =r => G = ab G = ±yfab 122 Note: (1) If a and b are positive then G = + ^[ab (2) If a and b are negative then G = - ^Jab (3) If a and b are opposite sign then their G.M is not real and it is discarded since we are dealing with real sequences. i.e. If a and b are opposite in signs, then G.M between them does not exist. Example 4. 7: Find n geometric means between two given numbers a and b and find their product. Solution: Let Gi, G2, • • • , G„ be n geometric means between a and b. By definition a, G\, G2, ■ • • , G n , b are in G.P. Let r be the common ratio. Then G\ = ar, G2 = ar , . . . , G n = ar n and b = ar 1 n + 1 Gi = a V\n+1 1 The product is V\n+1 1 G2 = a V\n+\ G w = a V\n + \ G\ . G2 . G3 . G n = a< V\n+1 IU\ n + 1 V\n+\ = a 1 +2+ ... +71 n+ 1 71(71 + 1)' V\ 2(72+1) a) n = (ab) 2 Example 4.8: Find 5 geometric means between 576 and 9. Solution: Let Gi, G2, G3, G4, G 5 be 5 G.Ms between a = 576 and b = 9 Let the common ratio be r Gi = 576r, G 2 = 576r 2 , G 3 = 576r 3 , G 4 = 576r 4 , G 5 = 576r 5 , 9 = 576r 6 123 r 6 = r = 9 576 J_ 2 '-{576J 6 -{64j 6 Gi =576r = 576x2 =288 G 3 = 576r 3 = 576 x | = 72 G 5 = 576r 5 = 576 x 32- = 18 G 2 = 576r 2 = 576 x \ = 144 G 4 = 576r 4 = 576 x jt = 36 4 J_ 16 Hence 288, 144, 72, 36, 18 are the required G.Ms between 576 and 9. Example 4.9: If b is the A.M of a and c (a ^ c) and (b - a) is the G.M of a and c - a, show that a : b : c = 1 : 3 : 5 Solution: Given b is the A.M of a and c .'. a, b, c are in A. P. Let the common difference be d :. b = a + d c = a + 2d Given (b - a) is the G.M of a and (c - a) :. (b - a) = a(c - a) 2 d = a(2d) => d=2a [:d*0] = a + d c = a + 2d = a + 2a c = a + 2(2a) ...(1) ...(2) From (1) and (2) = 3a c = 5a :. a : b : c = a : 3a : 5a =1:3:5 4.5.3 Harmonic mean H is called the harmonic mean between a and b if a, H, b are in H.P If a, H, b are in H.P then — , 77 , t are in A.P 1 H 1 1 a + b 1 _I I H "a + b 124 H = lab a + b This H is single H.M between a and b Definition: Hi, H2, ... H„ are called n harmonic means between a and b if a, Hi, H2, . . . H„, b are in H.P. Relation between A.M., G.M. and H.M. Example 4.10: If a, b are two different positive numbers then prove that (i) A.M., G.M., H.M. are in G.P. (ii) A.M > G.M > H.M Proof: a + lab A.M. = ^— ; G.M. = ^ab ; H.M. = -^ (1) G.M _ -Jab _ Isjab A.M ~~ a + b a + b 1 lab H.M a + G.M Isjab \[ab ~ a + b (1) ...(2) From (1) and (2) G.M _ H.M A.M ~ G.M .-. A.M, G.M, H.M are in G.P ,.. N . „ „ a + b i—r a + b-1-Jab (ii)A.M-G.M=— j— -^Jab = 2 — (cfa--fb) 2 = 2 >0 :a>0;b>0;a*b A.M > G.M ...(1) G.M-H.M = V^ -JTb _ ^[ab (a + b)- lab _ ^ab [a + b- l-\[ab] ~ a + b 2 > a + , ^[ab (^[a - ^jb) a + b :. G.M >H.M From (1) and (2) A.M. > G.M > H.M (2) 125 EXERCISE 4.2 (1) (i) Find five arithmetic means between 1 and 19 (ii) Find six arithmetic means between 3 and 17 (2) Find the single A.M between (i) 7 and 13 (ii) 5 and - 3 (iii) (p + q) and (p-q) (3) If b is the G.M of a and c and x is the A.M of a and b and y is the A.M a c of b and c, prove that — + - = 2 x y (4) The first and second terms of a H.P are -~ and t respectively, find the 9 th term. b + a b + c (5) If a, b, c are in H.P., prove that + •; = 2 r b - a b - c (6) The difference between two positive numbers is 18, and 4 times their G.M is equal to 5 times their H.M. Find the numbers. (7) If the A.M between two numbers is 1, prove that their H.M is the square of their G.M. 2 (8) If a, b, c are in A. P. and a, mb, c are in G.P then prove that a, m b,c are in H.P (9) If the p and q terms of a H.P are q and p respectively, show that (pq) 1 term is 1 . (10) Three numbers form a H.P. The sum of the numbers is 11 and the sum of the reciprocals is one. Find the numbers. 4.6 Some special types of series 4.6.1 Binomial series Binomial Theorem for a Rational Index: In the previous chapter we have already seen the Binomial expansion for a positive integral index n. (power is a positive integer) i , \1 n , /-i n— 11, /-in — 22, ,/->» — r r , ,-, n (x + a) =x + nL\ x a + nL-2X a +...+ nL r x a +... + nL n a A particular form is „ n(n - 1) 2 n ( n -!)("- 2 ) 3 „ (1 +xf = 1 +nx+ 2 , x z + — jf " jc j + ... +x n 126 When n is a positive integer the number of terms in the expansion is («+l) and so the series is a finite series. But when it is not a positive integer, the series does not terminate and it is an infinite series. Theorem (without proof) For any rational number n other than positive integer n n(n - 1) 2 , U(n -l)(n- 2) 3 (1 + x) = 1 + nx + r^ x + i j n x + provided I x I < I . Here we require the condition that I x I should be less than 1 . To see this, put x = 1 and n = - 1 in the above formula for (1 + x) n The left side of the formula = (1 + 1)~ = x , while the right side = 1 + (- 1) (1) + ( ~ ^ - l 2 + ... = 1-1 + 1-1 + ... Thus the two sides are not equal. This is because, x = 1 doesn't satisfy I x I < 1. This extra condition I x I < 1 is unnecessary, if n is a positive integer. Differences between the Binomial theorem for a positive integral index and for a rational index: 1. If n e N, then (1 + x) n is defined for all values of x and if n is a rational number other than the natural number, then (1 + x) n is defined only when I x I < I. 2. If n £ N, then the expansion of (1 + x) n contains only n + 1 terms. If n is a rational number other than natural number, then the expansion of (1 + x) n contains infinitely many terms. Some particular expansions We know that , when n is a rational index, n , \» 1 , , w( " ~ U 2 , n(n - 1) (n - 2) 3 (1+x) = 1 + nx + Tj x + Tj x + ... (1) Replacing x by - x, we get n-xr-l-»x + " ( "" 1) x 2 - »(^ -D(»- 2) 3 (2) y 1 A^ — 1 AtA T ^i A o | AT... v-^v Replacing w by - n in (1) we get (1 +x)~ n = 1 —nx + — 2\ — x 1 -— oj 1 L x 5 +... (3) n(n + 1) 2 »(» + 1) (n + 2) 3 127 Replacing x by - x in (3), we get ,, x-n , n ( n + 1) 2 »(" + 1) pi + 2) 3 (1-jc) "= 1 +/uc + - 2i — x + of x +... (4) Note: (1) If the exponent is negative then the value of the factors in the numerators are increasing uniformly by 1 (2) If the exponent is positive then the value of the factors in the numerators are decreasing uniformly by 1 (3) If the signs of x and n are same then all the terms in the expansion are positive. (4) If the signs of x and n are different, then the terms alternate in sign Special cases 1. (1 + x) _1 = 1 -x + x 2 -x 3 + ... 2. (1 -x)~ l = 1 +x + x 2 + x 3 + ... 3. (l+x)~ 2 = l-2x + 3x 2 -4x 3 + ... 4. (l-x)~ 2 = l+2x + 3x 2 + 4x 3 +... General term: For a rational number n and I x I < 1, we have „ n(n - 1) 2 , n(n - 1) (w - 2) 3 (1 + X) = 1 + MX + T^ X + 1 j -} x + • • • In this expansion First term Tj = Tq + 1 = 1 Second term T2 = Tj + \ = nx = j x n(n - 1) 7 Third term T3 = T2 + 1 = — p^ — x _ , w(w - 1) (n - 2) 3 Fourth term T4 = T3 + \ = TYt, x etc - , , n th t ^ n(n- 1) (n-2) ... (n-(r- 1)) r (r+1) term:T r+1 = ryt x The general term is n(n - 1) (n - 2) . . ,r factors r »(» - 1) (w - 2)...(n -r+1) r TY+ 1 = r i x = r | x Example 4.11: Write the first four terms in the expansions of (i)(l+4x)" 5 wherelxl<T (ii)(l-x 2 )" 4 wherelxl<l 128 Solution: (i) I 4x I = 41 x I < 4 I 4 j = 1 .-. I 4x I < 1 .". (1 + Ax) can be expanded by Binomial theorem. (5) (5 + 1) ,. , 2 (5) (5 + D (5 + 2) (1 + Ax)' J = 1 -( 5) {Ax) + y '\ 2 ' (Ax) 1.2.3 (4x) 3 +... = 1 - 20x + 15(16x 2 ) - 35(64x 3 ) + . . . = 1 - 20x + 2A0x 2 - 2240x 3 + . . . 9-4 2 (ii)(l-x) can be expanded by Binomial theorem since I x I < 1 , A r .w ?, (4) (4+1) , 2 2 ^ (4) (4+1) (4 + 2) 2 3 = 1 + (4) (x ) + j^2 (* ) + L23 ( x ) + • • • = 1 + 4x 2 + 10x 4 + 20x 6 + ... Example 4.12:Find the expansion of t where Ixl < 2 upto the fourth term. (2 + x) Solution: 1 (2 + x) ? =(2+x)" 4 = 2" 4 1+* \x\<2 < 1 J_ 16 J_ 16 l-(4) x\ , (4) (4 + 1) fx\ 2 (A) (4 + 1) (4 + 2) fx] 3 1.2 1.2.3 V2 1 -2x + " (4) (5) fx 2 ^ (4) (5) (6) . 2 U 1.2.3 8 + ... 16 8 + 32 x 32 x + Example 4.13:Show that (l+x) n = 2 n \-4 l-x\ fn+V\(l-x i+x) + n { 2! )(l + * ?♦...] _ . . T 1 -x Solution: Let y = , R.H.S = 2" " «(«+ 1) 2 1 l-/ry + 2 , y + ... = 2 n [l+y]-" = 2" r i-x" L 1 + i+*. - n = 2 n 1 +x+ 1 -x 1 +x - n = 2" ~ 2 " 1 +x_ - n = 2 n "1 +x 2 n = (l+x) n = L.H.S. 129 Approximation by using Binomial series Example 4.14: Find the value of ^126 correct to two decimal places. Solution: 1 1 3, ifl26 = (126) 3 = (125 + l) 3 1 1 J_Y 125 1 + 125/ 3 = (125) 3 | 1 i 1 _L I 3 125 J = 5 = 5 1 1 3 1 1 + 3 • 125 + 1 <1 + 3 (0.008) • 125 by neglecting other terms = 5[1+ 0.002666] = 5.01 (correct to 2 decimal places) Example 4.15: If x is large and positive show thaw.* + 6 - \]x + 3 =~~2 (app.) x Solution: Since x is large, ~ is small and hence < 1 ■\jx 3 + 6 --\jx 3 + 3 = (x 3 + 6) 3 - (x 3 + 3) 3 =x\l+— ] 3 -x\l+— ] 3 x 3 ) "TV 1 l 6 1 + 3.-3 +... •' X x + ^ + f... - X ^ + 5 3 ■7 3 + - - ; 1 + ... 2 ~x 2 " J X 2 (approximately) 1 Example 4.16: In the expansion (1 - 2x) 2 ) fj nc j the coefficient of jc . Solution: We know that (1-x) =l+nx+ — 21 — x +" »(n + 1) 2 »(» + 1) (n + 2) 3 n(n+l) ... (n + r-l) 3! X+...+ " r! -x +. General term T r + 1 = w(w + 1) ... (« + r- 1) Take n = 2 anc ^ re Pl ace x by 2x. 130 1 f3\f5\ (2r-\ t>.. iWW ? 2 W - 1 -"".-^-" ^ r r r! 2' " 1.3.5 ...(2r-l) . . coerncient or jc = ~j put r = 8 - 8 1.3.5.7.9.11.13.15 .•. coerncient or x = k\ 4.6.2. Exponential series Exponential theorem (without proof) For all real values of x, 11 1 Y x x 2 x 3 l+jy+27+...+^+...j = l+77 + 2! + 3! + 1 1 1 But e= 1 + 77 +T\ +T\ +••• 2 3 .". For all real values of x, e = 1+77+27+07 + Thus we have the following results: 2 3 >- A> •> •> e = 1 — YT + 2T — 3T + -- x , -x 2 4 e + e _ x_ x_ 2 ~ + 2! + 4! + ••• X —X e - e 2 3 5 X X = x+ 3! + 5! + ••• e + e 2 , 1 1 = 1+2! + 4! + " 2 1 1 1 _ 1! + 3! + 5! + ' 4.6.3 Logarithmic Series: 2 3 4 If - 1 < x < 1 then log(l + x)=x-^j~ + T" - "7" + This series is called the logarithmic series. 131 The other forms of logarithmic series are as follows: 2 3 log(l -x) = -x-~2 -~^ - ... 2 3 x x - log(l -x)=x + y+y+... / 3 5 X X log(l + X) - l0g(l - X) = 2 I X + ^ + "F + 1 1 +X X X 2 lo §r^ =x+ y + y + - EXERCISE 4.3 (1) Write the first four terms in the expansions of the following: (i) 7 where I x I > 2 (ii) where I x I < 2 (2 + x) 4 3r— — V6-3x (2) Evaluate the following: o (i) ^1003 correct to 2 places of decimals (ii) correct to 2 places of decimals i 2 1 — X X (3) If x is so small show that A / , = 1 - x + y (app.) (4) If x is so large prove that ^/x + 25 - Ajx +9 = ~ nearly. 11 (5) Find the 5 term in the expansion of (1 - 2x ) 2 (6) Find the (r + l) c term in the expansion of (1 - x) (7) Showthatx"=l+/l-£) + " ( " 2 1) (l-£f +... 132 5. ANALYTICAL GEOMETRY Introduction 'Geometry' is the study of points, lines, curves, surfaces etc and their properties. Geometry is based upon axioms and it was laid by the famous Greek Mathematician Euclid about 300 B.C. In the 17 c century A.D., the methods of Algebra were applied in the study of Geometry and thereby 'Analytical Geometry' emerged out. The renowned French philosopher and Mathematician Rene Descartes (1596 - 1650) showed how the methods of Algebra could be applied to the study of Geometry. He thus became the founder of Analytical Geometry (also called as Cartesian Geometry, from the latinized form of his name Cartesius). To bring a relationship between Algebra and Geometry, Descartes introduces basic algebraic entity 'number' to the basic geometric concept of 'point'. This relationship is called 'system of coordinates'. Descartes relates the position of a point with its distance from fixed lines and its direction. This chapter is a continuation of the study of the concepts of Analytical Geometry to which the students had been introduced in earlier classes. 5.1 Locus The path traced by a point when it moves according to specified geometrical conditions is called the locus of the point. For example, the locus of a point P(xj, y^) whose distance from a fixed point C (h, k) is constant 'a', is a circle (fig. 5.1). The fixed point 'C is called the centre and the fixed distance 'a' is called the radius of the circle. Example 5.1: A point in the plane moves so that its distance from (0, 1) is twice its distance from the x-axis. Find its locus. Solution: Let A(0, 1) be the given point. Let P(xj, jj) be any point on the locus. Let B be the foot of the perpendicular from P(x 1 ,y 1 ) to the x-axis. Thus PB =y^. Given that PA = 2PB P(*i,yi) Fig. 5. 1 YT A(0, 1) i.e. .-. PA 2 = 4PB 2 (x l -0) 2 + (y 1 -l) 2 = 4y l 2 O P(*i,yi) EL B Fig. 5. 2 133 2 2 2 i.e. Xj +)>j -2y l + l=4y l i.e. JCj 2 - 3jj 2 - 2y l + 1 = 2 2 .". The locus of (Xp jj) is x - 3j -2j + 1 = Example 5.2: Find the locus of the point which is equidistant from (- 1, 1) and (4, - 2). Solution: Let A(— 1,1) and B(4, - 2) be the given points. Let P(x 1 ,y 1 ) be any point on the locus. Given that PA = PB .-. PA 2 = PB 2 i.e. (xj + 1) 2 + (y x - l) 2 = (x t - 4) 2 + (^ + 2) 2 i.e. Xj + 2x 1 + l+;y 1 -2y 1 + l=x 1 -8x^ + 16 + )^ + 4^+4 i.e. lObCj - 6y x - 18 = i.e. 5x 1 -3 y x - 9 = .". The locus of the point (xj, yj) is 5x - 3y - 9 = Example 5.3: If A and B are the two points (- 2, 3) and (4, - 5), find the 2 2 equation of the locus of a point such that PA - PB = 20. Solution: A(— 2, 3) and B(4, - 5) are the two given points. Let P(xj, y^) be any point on the locus. Given that PA 2 - PB 2 = 20. (xj + 2) 2 + (y 1 - 3) 2 - [(xi- 4) 2 + (yi+5) 2 ] = 20 x 1 2 + 4x 1 +4+;y 1 2 -6;y 1 + 9-[x 1 2 -8x 1 + 16 + ;y 1 2 +10;y 1 + 25] = 20 12*!- 16^-48 = i.e. 3xj - 4y l - 12 = The locus of (Xpjj) is 3x - Ay - 12 = Example 5.4: Find a point on x-axis which is equidistant from the points (7, - 6) and (3, 4) . Solution: Let P(xj, y 1 ) be the required point. Since P lies on x-axis, y^ = 0. Given that A(7, - 6) and B(3, 4) are equidistant from P. i.e. PA = PB => PA 2 = PB 2 => (xj - 7) 2 + (0 + 6) 2 = (xj - 3) 2 + (0 - 4) 2 134 => xj 2 - 14xj + 49 + 36 = xj 2 - 6x l + 9 + 16 => 8xj = 60 .-. x 1 = 15/2 Thus the required point is [pr ,0 EXERCISE 5.1 (1) A point moves so that it is always at a distance of 6 units from the point (1, - 4). Find its locus. (2) Find the equation of the locus of the point which are equidistant from (1,4) and (-2, 3). (3) If the point P(5f - 4, t + 1) lies on the line 7x - Ay + 1 = 0, find (i) the value of t (ii) the co-ordinates of P. (4) The distance of a point from the origin is five times its distance from the y-axis. Find the equation of the locus. (5) Show that the equation of the locus of a point which moves such that its distance from the points (1, 2) and (0,-1) are in the ratio 2 : 1 is 3x + 3y 2 + 2x+ 12j-l=0. (6) A point P moves such that P and the points (2, 3), (1, 5) are always collinear. Show that the equation of the locus of P is 2x + y - 7 = 0. (7) A and B are two points (1, 0) and (- 2, 3). Find the equation of the locus of a point such that (i) PA 2 + PB 2 = 10 (ii) PA = 4PB. 5.2 Straight lines 5.2.1 Introduction A straight line is the simplest geometrical curve. Every straight line is associated with an equation. To determine the equation of a straight line, two conditions are required. We have derived the equation of a straight line in different forms in the earlier classes. They are (1) Slope-intercept form: i.e. y = mx + c where 'm' is the slope of the straight line and V is the y intercept. (2) Point-slope form: i.e. y — yj = m(x - x^) where c m' is the slope and (xp y^) is the given point. (3) Two point form: y-y\ x-xi i.e = where (x,, v,) and (x , y ) are the two given points. y2~yi X2-X1 i J1 ' 2;2 135 (4) Intercept form: x y i.e. — + f = 1 where 'a' and 'b' axe x and y intercepts respectively. In this section we shall derive and discuss other forms of equation of a straight line. 5.2.2 Normal form: Equation of a straight line in terms of the length of the perpendicular p from the origin to the line and the angle a which the perpendicular makes with jc-axis. Let R and N be the points where the straight line cuts the x and y axes respectively. Draw the perpendicular OL to RN. Let OL = p and |XOL = a. Now OR and ON are the x and y intercepts respectively. Fig. 5. 3 x y The equation of the straight line is t^t +7y\T =1 •••(!) OR From the right angled triangle OLR, sec a = 7^~ .'. OR = p sec a ON From the right angled triangle OLN, cosec a = sec (90 - a) = j^r .". ON =p coseca Substituting the values of OR and ON in equation (1), x y x cos a y sin a we get, + = 1 i.e. — - — + — - — = 1 p sec a p cosec a P P i.e. x cos a + y sin a = p is the required equation of the straight line. 5.2.3 Parametric form Definition: If two variables, say x and y, are functions of a third variable, say '9', then the functions expressing x and y in terms of 9 are called the parametric representations of x and y. The variable 9 is called the parameter of the function. Equation of a straight line passing through the point (x v jj) and making an angle 9 with x-axis. (parametric form) 136 Let Q (xj, yj) be the given point and P(x, y) be any point on the required straight line. Assume that PQ = r. It is given that |PTR = 9. But |PQM = |PTR .-. |PQM = 9 In the right angled triangle PQM, ...(1) QM NR OR -ON x-x { v ■■ i 1 Q x % M A§ n "1 fx o N K Fig. 5. 4 cos9 = PQ x — xi cos9 = r n PM PR - MR Similarly sin9 = p7j = ~ y-y\ y-y\ sin9 = r (2) From (1) and (2), x-x\ y-y\ ~z = — r~r- = r which is the required equation. cos Any point on this line can be taken as (x\ + r cos 9, y\ + r sin 9) where r is the algebraic distance. Here r is the parameter. 5.2.4 General form The equation ax + by + c = will always represent a straight line. Let (x,, y,), (x 2 , y 2 ) anc ^ ^ x 3' ^3) ^ e an y triree points on the locus represented by the equation ax + by + c = 0. Then ax, + by. + c = ••■(!) ax, + by 2 + c = . . . (2) ax 3 + by 3 + c = . . . (3) (1) x (y 2 - y 3 ) + (2) x (y 3 - yj + (3) x(y l ~ y 2 ) gives a [x[ (yi - yj) + x 2 O3 - yi) + x 3 (yi - y 2 )] = Since a * 0, Xj (y 2 - y 3 ) + x 2 (y 3 - yj + x 3 (y t - y 2 ) = 137 That is (xj, y^), (x 2 , y 2 ) anc ^ ( x 3' ^3) are collinear and hence they lie on a straight line. Thus the equation ax + by + c = represents a straight line. 5.2.5. Perpendicular distance from a point to a straight line The length of the perpendicular from the point (xj,jj) to the line ax\ + by\ + c ax + by + c = is (1) V^7 2 Let the given line ax + by + c = be represented by AB. Let P(xj, jj) be the given point. Draw PD perpendicular to AB. Note that PDis the required distance. x Draw OM parallel to PD. Let OM = p Assume that |MOB = a. Fig. 5. 5 From 5.2.2, the equation of the straight line AB is x cosa + y sina - p = . . . (2) Now equations (1) and (2) are representing the same straight line. Hence their corresponding coefficients are proportional. cos a sina - p a ~ b ~ c ap . pb cosa = — , sina = - ~~ We know that 2 2 sin a + cos a = 1 2, 2 2 2 p b pa 2 2, 2,2 = 1 i.e. pa + p b = c 2^ 2 , , 2, 2 ■ 2 c p (a +b ) = c i.e p = 2 ,2 a + b Hence cosa = = + ^[a 2 + b 2 , sina = Va 2 + b 2 b \]a 2 + b 2 138 Suppose OL = p', the equation of the straight line NR is x cos a + y sin a - p' = since P(Xp y^) is a point on NR Xj cos a + y l sin a - /?' = i.e. OL =p' =x l cos a + y^ sin a From the figure, the required distance PD = LM = OM - OL =p -p' = p - x, cosa - y, sin a c x\ . a y\ . b ax\+byi+c ~ ^ja 2 + b 2 ~ \ja 2 + b 2 ^ja 2 + b 2 ~ -\ja 2 + b 2 ax\ + by i + c The required distance = \[a 2 + b 2 Corollary: The length of the perpendicular from the origin to ax + by + c = is c la 2 + b 2 c 4 Note: The general equation of the straight line is ax+by+c = i.e. y = -Tx - ~ This is of the form y = mx + c. a . co-efficient of x :. m = - B i.e. slope = - co . efficient of " Example 5.5: Determine the equation of the straight line whose slope is 2 and j-intercept is 7. Solution: The slope - intercept form is y = mx + c Here m = 2, c = 7 .". The required equation of the straight line is y = 2x + 7 Example 5.6: Determine the equation of the straight line passing through 2 (- 1,2) and having slope j Solution: The point-slope form isy -y l = m(x - x^). 2 Here (xj, y^) = (- 1, 2) and m = ij 2 y-2 = j(x+l) i.e. ly - 14 = 2x + 2 2x - ly + 16 = is the equation of the straight line. 139 Example 5. 7: Determine the equation of the straight line passing through the points (1,2) and (3, -4). Solution: y-yi x-x\ The equation of a straight line passing through two points is = y\-yi x\-X2 Here (x v y { ) = (1, 2) and (x 2 , y 2 ) = (3, - 4). y — 2 x — 1 Substituting the above, the required line is 9 + 4 = 1 — T y - 2 x - 1 y - 2 x - 1 ^ 6 = -2 ^ 3 = -1 =^> y-2 = -3(x-l)^>y-2 = -3x + 3 => 3x + y = 5 is the required equation of the straight line. Example 5.8: Find the equation of the straight line passing through the point (1, 2) and making intercepts on the co-ordinate axes which are in the ratio 2:3. Solution: x y The intercept form is — +f = 1 ■■■(I) The intercepts are in the ratio 2:3 .'. a = 2k, b = 3k. x y (1) becomes jT + ~ti =i x - e - 3x + 2y = 6k Since (1, 2) lies on the above straight line, 3 + 4 = 6k i.e. 6k = 7 Hence the required equation of the straight line is 3x + 2y = 7 Example 5.9: Find the length of the perpendicular from (2, - 3) to the line 2x - y + 9 = Solution: The perpendicular distance from (jcj, y^) to the straight line ax + by + c = ax\ + by\ + c is given by V« 2 , j2 + b ..The length of the perpendicular from (2, -3) to the straight line 2(2) - (- 3) + 9 2x - y + 9 = is V(2) 2 + (-D 2 16 ■■^ units. Example 5.10: Find the co-ordinates of the points on the straight line y = x + 1 which are at a distance of 5 units from the straight line 4x - 3y + 20 = 140 Solution: Let (xj, y^) be a point on y = x + 1 :.y l =x 1 + l ...(1) The length of the perpendicular from (xp y^) to the straight line 4x - 3y + 20 = is 4xi - 3yi + 20 , 4xi - 3yi + 20 V4 2 + (- 3) 2 But the length of the perpendicular is given as 5. ^4xi - 3yi + 20 5 ' = 5 :. 4xj - 3y l + 20 = + 25 Considering the positive sign, 4xj - 3y^ + 20 = 25 => 4x l -3y l = 5 ... (2) Considering the negative sign, 4xj - 3yj + 20 = - 25 => 4x x - 3y x =-45 ...(3) Solving (1) and (2), we get x 1 = 8, yj = 9 Solving (1) and (3), we get Xj=-42, ^=-41. .". The co-ordinates of the required points are (8, 9) and (- 42, - 41). Example 5.11: Find the equation of the straight line, if the perpendicular from the origin makes an angle of 120° with x-axis and the length of the perpendicular from the origin is 6 units. Solution: The normal form of a straight line is x cosa + y sina = p Here a = 120°, p = 6 :. x cos 120° + y sin 120° = 6 => x i~2J +y { 2 J = 6 ^ -x + y{3y= 12 => x-V3y+12 = .". The required equation of the straight line is x - y[3 y + 12 = Example 5.12: Find the points on y-axis whose perpendicular distance from the straight line 4x - 3y - 12 = is 3. Solution: Any point on j-axis will have x co-ordinate as 0. Let the point on y-axis be P(0, y^). 141 The given straight line is 4x - 3j - 12 = • • • (1) The perpendicular distance from the point P to the given straight line is 3yi - 12 3yi + 12 \J4 2 + (- 3) 2 5 But the perpendicular distance is 3. i.e. 3yi + 12 5 = 3 => 3y t + 12 = +15 3y 1 + 12 = 15 or 3y t + 12 = - 15 3y 1= 3 or 3^ =-27 ) 1 = 1 or >'!=-' ; Thus the required points are (0, 1) and (0, - 9). EXERCISE 5.2 (1) Determine the equation of the straight line passing through the point 4 (-1,-2) and having slope 7 (2) Determine the equation of the line with slope 3 and j-intercept 4. (3) A straight line makes an angle of 45° with x-axis and passes through the point (3, - 3). Find its equation. (4) Find the equation of the straight line joining the points (3, 6) and (2,-5). (5) Find the equation of the straight line passing through the point (2, 2) and having intercepts whose sum is 9. (6) Find the equation of the straight line whose intercept on the x-axis is 3 times its intercept on the j-axis and which passes through the point (-1,3). (7) Find the equations of the medians of the triangle formed by the points (2, 4), (4, 6) and (-6,-10). (8) Find the length of the perpendicular from (3, 2) to the straight line 3x + 2y+ 1=0. (9) The portion of a straight line between the axes is bisected at the point (- 3, 2). Find its equation. (10) Find the equation of the diagonals of a quadrilateral whose vertices are (1,2), (-2,-1), (3, 6) and (6, 8). 142 (11) Find the equation of the straight line, which cut off intercepts on the axes whose sum and product are 1 and - 6 respectively. (12) Find the intercepts made by the line Ix + 3y - 6 = on the co-ordinate axis. (13) What are the points on x-axis whose perpendicular distance from the x y straight line ^ + J = 1 is 4? (14) Find the distance of the line 4x - y = from the point (4, 1) measured along the straight line making an angle of 135° with the positive direction of the x-axis. 5.3. Family of straight lines In the previous section, we studied about a single straight line. In this section we will discuss the profile about more than one straight line, which lie on a plane. 5.3.1 Angle between two straight lines Let L:y = m*x + c, and y Z 2 : y = m^x + c 2 be the two intersecting lines and assume that P be the point of intersection of the two straight lines which makes angle 9j and 9 2 with the positive direction of x-axis. Then m l = tan9j and m 2 = tan9 2 - Let 9 be the angle between the two straight lines. Fig. 5. 6 From the figure (5.6), 9j = 9 + 9 2 .'. 9 = 9,— 9^ tan9 = tan (9j - 9 2 ) - m.2 tan9i - tan92 m\—m.2 1 + tan0 1 .tan0 2 \+m\m2 mi ~ m 2 Note that f" ; is either positive or negative. As convention we 1 + m\ mi v b consider the acute angle as the angle between any two straight lines and hence we consider only the positive value (absolute value) of tan9. Hence tan9 = m\ — ni2 1 + m\ m.2 9 = tan m\ — m.2 1 + m\ m.2 143 Corollary (1) : If the two straight lines are parallel, then their slopes are equal. Proof: Since the two straight lines are parallel, 9 = 0. .'. tan 9 = m.\ - m.2 => "T~ =0 => m. -m~ = 1 + mi m.2 12 i.e. m, = niy :. If the straight lines are parallel, then the slopes are equal. Note : If the slopes are equal, then the straight lines are parallel. Corollary (2): If the two straight lines are perpendicular then the product of their slopes is -1. Proof: Since the two straight lines are perpendicular, 9 = 90°. m\ — m.2 :. tanO = tan90° = oo => , = co 1 + m\ m2 This is possible only if the denominator is zero. i.e. 1 + m, m 2 = i.e. m, m 2 = - 1 .". If the two straight lines are perpendicular then the product of their slopes is - 1 . Note (1): If the product of the slopes is - 1, then the straight lines are perpendicular. (2): Corollary (2) is applicable only if both the slopes m^ and m 2 are finite. It fails when the straight lines are co-ordinate axes or parallel to axes. Corollary (3): If the straight lines are parallel, then the coefficients of x and y are proportional in their equations. In particular, the equations of two parallel straight lines differ only by the constant term. Proof: Let the straight lines a^x + b^y + Cj = and a 2 x + b 2 y + c 2 = be parallel. a\ Slope of a,x + b,y + c, = is m, = - ~r~ ; Slope of a 2 x + b 2 y + c 2 = is (22 m 2=-V 2 Since the straight lines are parallel, m l = m 2 . a\ a.2 a\ b\ Le ' ~ b\ ~ ~ b2 ^ ci2 b2 144 i.e. coefficients of x and y are proportional Let tt, = h = x ^ :. a 2 = a,X, b 2 = b, X The second equation a 2 x + b 7 y + c 2 = can be written as Xa , x + X b , y + c 2 = c 2 . c 2 i.e. a,x + b,y + t~ =0 i.e. a-,x + b,y + k = where k = ~r i.e. If a^x + b^y + c^ = is a straight line then a line parallel to it is a,x + b,y + k = :. Equations of parallel straight lines differ by the constant term. Note (1): In the previous section, we established a formula to find the distance between the origin and the straight line. i.e. distance = "V^ 2 + b 2 We can find out the distance between two parallel straight lines I c\ —ci I ax + by + Cj = and ax + by + c 2 = by using the formula d = , . ~\ja + b This is obtained by using the above result. Note that, we took I Cj - c 2 I since Cy > c, or c, > c 2 Note (2): To apply the above formula, write the equations of the parallel straight lines in the standard form ax + by + c l = and ax + by + c 2 = 0. Corollary (4): The equation of the straight line perpendicular to the straight line ax + by + c = is of the form bx - ay + k = for some k. Proof: Let the straight lines ax + by + c = and a^x + b^y + c^ = be perpendicular. a Slope of ax + by + c = is m^ = - t a\ Slope a,x + b.y + c, = is m 2 = — ~r~ Since the straight lines are perpendicular, m^m 2 = -\ i.e. -T7 -T =-1 i.e. aa, = -bb. l \ 145 Cl\ i.e. = X (say) .". a, = bX and b,=-aX The second equation ajX + ^y + Cj =0 can be written as bXx - aXy + c^ = i.e. i.e. bx- ay + ~r = bx- ay + k = where fc = £1 A straight line perpendicular to ax + by + c = is given by bx - ay + k = for some k. Note: To find the point of intersection of two straight lines, solve the simultaneous equations of the straight lines. 5.3.3 The condition for the three straight lines to be concurrent Let the three straight lines be given by ayX + b^y + Cj = ... (1) a 2 x + b 2 y + c 2 = . . . (2) a^x + b^y + c 3 = . . . (3) If the three straight lines are concurrent, then the point of intersection of any two straight lines lies on the third straight line. Solving the equation (1) and (2), the coordinates of the point of intersection is b\ c 2 -b 2 c\ c\a 2 -c 2 a\ x — v — a\b 2 - a 2 b\ a\b 2 - a 2 b\ substituting the values of x and y in the equation (3) b\ c 2 - b 2 c\ 3\a\b 2 - a 2 b[) ^\a\b 2 — a 2 b\ c\a 2 -c 2 a\ + c 3 = i.e. <au Q ) \ c 2 ~ b 2 c,) +bJc,a 2 - c 2 aS) + c-AaJjy ~ <*J>\) = i.e. a^b^c^ - b^c 2 ) - b^(a 2 c^ a\ b\ c\ 3 c 2 ) + c^(a 2 b^ - a^b 2 ) = i.e. a 2 b 2 c 2 aj, b 3 c 3 concurrent. = is the condition for the three straight lines to be 146 5.3.4 Equation of a straight line passing through the intersection of the two given straight lines Let a ] x + b l y + c^ = a 2 x + b 2 y + c 2 = (1) (2) (3) be the equations of the two given straight lines. Consider the equation a^x + b^y + c^ + X (a 2 x + b 2 y + c 2 ) = where X is a constant Equation (3) is of degree one in x and y and therefore (refer 5.2.4) it represents a straight line. Let (x { , y^) be the point of intersection of (1) and (2) .". a,x, + b^y, + c, = and a 2 x, + b 2 y, + c 2 = .'. a,x, + b,y, + c, + X (a 2 x. + b 2 y, + c 2 ) = .". Value of (x-p y^) satisfies equation (3) also. Hence a^x + b^y + c^ + X (a 2 x + b 2 y + c 2 ) = represents a straight line passing through the intersection of the straight lines a^x + b { y + c^ = and a,x + b 2 y + c 2 = Example 5.13: Find the angle between the straight lines 3x - 2y + 9 = and 2x + y - 9 = 0. Solution: 3 9' ■■y=2 x+ 2 Slope of the straight line 3x - 2y + 9 = is m.y =j Slope of the straight line 2x + y - 9 = is m 2 = - 2 [ v y = - 2x + 9] Suppose '9' is the angle between the given lines, then -1 9 = tan = tan = tan 1 + m\ m.2 2 + 1 l+|(-2) = tan 7 2 2-6 = tan Example 5.14: Show that the straight lines 2x + y - 9 = and 2x + y - 10 = are parallel. 147 Solution: Slope of the straight line 2x + y-9 = 0ism^ = -2 Slope of the straight line 2x + y - 10 = is m 2 = - 2 :. m l = m 2 .'. The given straight lines are parallel. Example 5.15: Show that the two straight lines whose equations are x + 2y + 5 = and 2x + Ay - 5 = are parallel. Solution: The two given equations are x + 2y + 5 = 2x + Ay - 5 = 1 2 The coefficients of x and y are proportional since ~ = 7 and therefore they are parallel. Note : This can also be done by writing the equation(2) as x + 2y - 5/2 = Now the two equations differ by constant alone. .". They are parallel. Example 5.16: Find the distance between the parallel lines 2x + 3y - 6=0 and 2x + 3y + 1 = 0. Solution: C1-C2 (1) (2) The distance between the parallel lines is Here Cj = - 6, c 2 = 7, a = 2, b = 3 V a 2 + b 2 The required distance is -6-7 V2 2 + 3 2 = -13 Vl3 = ^/l3 units. Example 5.17: Show that the straight lines 2x + 3y - 9 = and 3x - 2y + 10 = are at right angles. Solution: Slope of the straight line Slope of the straight line 2x + 3y - 9 = is Wj = - ^ 3 3x - 2>> + 10 = is m 2 = 7 2 3 . . m,m 2 = - 7 -7 = - 1 The two straight lines are at right angles. 148 Example 5.18: Find the equation of the straight line parallel to 3x + 2y = 9 and which passes through the point (3, - 3). Solution: The straight line parallel to 3x + 2y - 9 = is of the form 3x + 2y + k = ... (1) The point (3, - 3) satisfies the equation (1) Hence 9 - 6 + k = i.e. k = - 3 .". 3x + 2y - 3 = is the equation of the required straight line. Example 5. 19: Find the equation of the straight line perpendicular to the straight line 3x + Ay + 28 = and passing through the point (- 1, 4). Solution: The equation of any straight line perpendicular to 3x + Ay + 28 = is of the form Ax - 3y + k = The point (- 1,4) lies on the straight line Ax - 3y + k = .-. -4-12 + fe = => k = 16 .". The equation of the required straight line is Ax - 3y + 16 = Example 5. 20: Show that the triangle formed by straight lines Ax - 3y - 18 = 0, 3x - Ay + 16 = and x + y-2 = 0is isosceles. Solution: Slope of the straight line Ax - 3y ■ 18 = is m,y = t Slope of the straight line 3x - Ay + 16 = is m 2 = T Slope of the straight line x + >>-2 = 0ism 3 = - 1 Let 'a' be the angle between the straight lines Ax 3x- Ay +16 = tri[ -mi Using the formula, 9 = tan 1 + m\ OT2 we get a = tan = tan = tan 24 3y - 18 = and 4 3 16-9 3"4 , 4 3 1 + 34 = tan 1 12 2 149 Let '(3' be the angle between the straight lines 3x -4y+l6 = and x + y - 2 = 7/4 (3 = tan" 1 !♦■ i+l^" 1 ) = tan ■l 1/4 = tan" 1 (7) Let 'y' be the angle between the straight lines x+y -2 = and 4x-3y - 18= .". y = tan l + (-Dl3 = tan = tan" 1 (7) Therefore (3 = y •'• The triangle is isosceles. Example 5.2i:Find the point of intersection of the straight lines 5x + Ay - 13 = and 3x + y - 5 = Solution: To find the point of intersection, solve the given equations. Let (xj, jj) be the point of intersection. Then (jci, y\) lies on both the straight lines. .-. 5x 1 +4y 1 = 13 ...(1) 3^+^ = 5 ...(2) (2)x4 ^ 12x^4^=20 ...(3) (l)-(3) =$ -lx x =-7 .-. x x = \ Substituting x^ = 1 in equation (1), we get 5 + Ay^ = 13 Ay x =8 .". y, = 2 The point of intersection is (1, 2). Example 5.22: Find the equation of the straight line passing through the intersection of the straight lines 2x + y = 8 and 3x - y = 2 and through the point (2,-3) Solution: The equation of the straight line passing through the intersection of the given lines is 2x + y-8 + X(3x-y-2) = ...(1) (2, - 3) lies on the equation (1) and hence A-3 - 8 + X (6 + 3- 2) = 150 .-. X = 1 .-. (1) => 2x + y-8 + 3x-y-2 = => 5jc-10 = x = 2 is the equation of the required straight line. Example 5.23: Find the equation of the straight line passing through the intersection of the straight lines 2x + y = 8 and 3x - 2y + 7 = and parallel to Ax + y-ll=0 Solution: Let (xp jj) be the point of intersection of the given straight lines 2xy + y l = 8 ...(1) 3xj - 2jj = - 7 ...(2) (l)x2 => 4xj + 2y l = 16 ...(3) (2) + (3) => 9 ■'■ x \ ~ 1 38 (9 38^ y 1= 7 .. (x 1 ,y 1 ) = ( J , 7 J The straight line parallel to 4x + y - 1 1 = is of the form Ax + y + k = But it passes through \j ,y 1 36 38 ■y +^y" +fc = • k -- 7 4 74 4jc + j - -y =0 28x + ly - 1A = is the equation of the required straight line. Example 5.24: Find the equation of the straight line which passes through the intersection of the straight lines 5x - 6y = 1 and 3x + 2y + 5 = and is perpendicular to the straight line 3x - 5y + 1 1 = Solution: The straight line passing through the intersection of the given straight lines is 5x-6y-l+X(3x + 2y + 5) = ...(1) (5 + 3X)x + (- 6 + 2X)y + (- 1 + 5X) = This straight line is perpendicular to 3x - 5y + 1 1 = Product of the slopes of the perpendicular straight lines is -1 i.e. ?Mj m 2 = -1 5 + 3^ ^ (?>, 6 + 2XJ \5; 15 + 9?. = -30+m .-.A. = 45 151 (1) => 5x - 6y - 1 + 45 (3x + 2y + 5) = i.e. UOx + My + 224 = i.e. 5x + 3y + 8 = is the equation of the required straight line. Example 5.25: Show that the straight lines 3x + 4y = 13; 2x - ly + 1=0 and 5x - y = 14 are concurrent. Solution: Let (xj, jj) be the point of intersection of the first two straight lines 3*1 + %i 2xi " 7 ?l (l)x7 (2)x4 (3) + (4) (1) 2lx l + 28y l = 8xj - 28^ = 29xj = 13 ...(1) -1 ...(2) 91 ...(3) -4 ...(4) 87 => jcj = 3 13 => y, =1 => 9 + Ay x The point of intersection of the first two straight lines is (3, 1). Substitute this value in the equation 5x - y = 14 L.H.S. = 5x-y = 15 - 1 = 14 = R.H.S. i.e. The point (3, 1) satisfies the third equation. Hence the three straight lines are concurrent. Example 5.26: Find the co-ordinates of orthocentre of the triangle formed by the straight lines j C - } ;-5 = 0, 2x-y-8 = 0and3jc-)'-9 = Solution: Let the equations of sides AB, BC and CA of a AABC be represented by X -y-5 = ...(1) 2x-y-8 = ...(2) 3x-y-9 = ...(3) Solving (1) and (3), we get A as (2, - 3) B The equation of the straight line BC is 2x perpendicular to it is of the form x + 2y + k = D 2x-y-S = Fig. 5. 7 8 = 0. The straight line ...(4) 152 A(2, - 3) satisfies the equation (4) .-. 2 - 6 + k = =^> A; = 4 The equation of AD is x + 2y = - 4 . . . (5) Solving the equations (1) and (2), we get B as (3, - 2) The straight line perpendicular to 3x - y - 9 = is of the form x + 3y + k = But B(3, - 2) lies on this straight line :. 3-6 + k = => k = 3 .". The equation of BE is x + 3y = - 3 . . . (6) Solving (5) and (6), we get the orthocentre O as (- 6, 1). Example 5.27: For what values of 'a', the three straight lines 3x + y + 2 = 0, 2x - y + 3 = and x + ay - 3 = are concurrent? Solution: Let (xp yj) be the point of concurrency. This point satisfies the first two equations. .-. 3x 1 +y 1 + 2 = ...(1) 2xj-yj + 3 = ...(2) Solving (1) and (2) we get (- 1, 1) as the point of intersection. Since it is a point of concurrency, it lies on x + ay - 3 = 0. .-. - 1 + a - 3 = i.e. a = 4 EXERCISE 5.3 (1) Find the angle between the straight lines 2x + y = 4 and x + 3y = 5 (2) Show that the straight lines 2x + y = 5 and x - 2y = 4 are at right angles. (3) Find the equation of the straight line passing through the point (1, - 2) and parallel to the straight line 3x + 2y - 7 = (4) Find the equation of the straight line passing through the point (2, 1) and perpendicular to the straight line x + y = 9 (5) Find the point of intersection of the straight lines 5x + 4y - 13 = and 3x + y - 5 = (6) If the two straight lines 2x - 3y + 9 = 0, 6x + ky + 4 = are parallel, find k (7) Find the distance between the parallel lines 2x + y - 9 = and 4x + 2y + 7 = (8) Find the values of p for which the straight lines 8px + (2 - 3p) y + 1 = and px + 8y - 7 = are perpendicular to each other. 153 (9 (10 (11 (12 (13 (14 (15 (16 (17 (18 (19 (20 (21 (22 Find the equation of the straight line which passes through the intersection of the straight lines 2x + y = 8 and 3x - 2y + 7 = and is parallel to the straight line Ax + y - 11 = Find the equation of the straight line passing through intersection of the straight lines 5x - 6y = 1 and 3x + 2y + 5 = and perpendicular to the straight line 3x - 5y + 1 1 = Find the equation of the straight line joining (A, - 3) and the intersection of the straight lines 2x - y + 7 = and x + y - 1 = Find the equation of the straight line joining the point of the intersection of the straight lines 3x + 2y + 1 = and x + y = 3 to the point of intersection of the straight lines y — x = 1 and 2x + y + 2 = Show that the angle between 3x + 2y = and Ax - y = is equal to the angle between 2x + y = and 9x + 32y = 41 Show that the triangle whose sides are y = 2x + 7, x - 3y - 6 = and x + 2y = 8 is right angled. Find its other angles. Show that the straight lines 3x + y + A = 0, 3x + Ay - 15 = and 24x - ly - 3 = form an isosceles triangle. Show that the straight lines 3x + 4>>=13; 2x - ly + 1 = and 5x - y = 14 are concurrent. Find 'a' so that the straight lines x - 6y + a = 0, 2x + 3y + A = and x + Ay + 1 = may be concurrent. Find the value of 'a' for which the straight lines x + y-A = 0,3x + 2 = and x - y + 3a = are concurrent. Find the co-ordinates of the orthocentre of the triangle whose vertices are the points (- 2, - 1), (6, - 1) and (2, 5) If ax + by + c = 0, bx + cy + a = and ex + ay + b = are concurrent, show that a 3 + b 3 + c 3 = 3abc Find the co-ordinates of the orthocentre of the triangle formed by the straight lines x + y - I =0, x + 2y - 4 = and x + 3y - 9 = The equation of the sides of a triangle are x + 2y = 0, Ax + 3y = 5 and 3x + y = 0. Find the co-ordinates of the orthocentre of the triangle. 5.4 Pair of straight lines 5.4.1 Combined equation of the pair of straight lines We know that any equation of first degree in x and y represents a straight line. Let l^x + m^y + n^ = and l 2 x + m 2 y + n 2 = be the individual equations of any two straight lines. Then their combined equation is (Lx + m.y + «,) (Lx + m^y + « 2 ) = 154 2 2 ZjZ 2 -x + (/jm 2 + /2 OT i) *y + m^rn^y +(1^2 + / 2 M i) x + (tn^ + m^n^y + n^x^ = Hence the equation of a pair of straight lines may be taken in the form 2 2 ax + 2hxy + by + 2gx + 2fy + c = 0, where a, b, c,f, g, h are constants. 5.4.2 Pair of straight lines passing through the origin 2 2 The homogeneous equation ax + 2hxy + by = of second degree in x and y represents a pair of straight lines passing through the origin. 2 2 Considering ax + 2hxy + by = as a quadratic equation in x, we get / 2 2 2 -2hy ±-\J4 h y -Aaby la -2h± 2-\jh 2 - ab 2a y h ± \lh 2 - ab y = (-h±^h 2 -ab) y i.e. ax+\h+ \]h - ab) y = and ax + \h- 'Sjh - ab) y = are the 2 2 two straight lines, each passing through the origin. Hence ax + 2hxy + by = represents a pair of straight lines intersecting at the origin. 2 Note : The straight lines are (1) real and distinct if h > ab 2 (2) coincident if h = ab 2 (3) imaginary if h <ab Sum and product of the slopes of pair of straight lines 2 2 The homogeneous equation ax + 2hxy + by = of second degree in x and y represents a pair of straight lines passing through the origin. Let y = m^x and y = m^x be the two straight lines passing through the origin. Therefore the combined equation is (y — m^x) (y - m^x) = 2 2 => mj/fiji -(m^+m^xy + y =0 This equation also represents a pair of straight lines passing through the origin. Equating the co-efficients of like terms in the above equations, we get m\m.2 (m\ + mj) \ a ~ 2h ~ b a a :. m,y m 2 = r ; i.e. Product of the slopes = t 2h . „ r , , 2h m,y + m 2 = —~i~ i.e. Sum of the slopes = -~r 155 5.4.3 Angle between pair of straight lines passing through the origin The equation of the pair of straight lines passing through the origin is 2 2 ax + 2hxy + by = 1h a m.+m2 = —~iT and m. m 2 = r Let '9' be the angle between the pair of straight lines. m\ - m.2 (1) tanG = tan 9 = 1 + m\ m.2 \j(m[ + m.2) -4m[tn2 1 + otj m.2 \ \hf- 4a b 2 ~ b a l+r 4h z - 4ab a + 1 tan 9 = + 2\jh 2 - ab a + b 9 = tan" 1 ±2-\jh 2 a + 1 - ab » i.e. 9 = tan ■d 2\h^-ab a + 1 It is conventional to take 9 to be acute. Corollary (1): If '9' is the angle between the pair of straight lines 2 2 ax + 2hxy + by + 2gx + 2fy + c = then 9 = tan 2 -\jh 2 - ab a + b It is same as the angle between the pair of straight lines 2 2 ax + 2hxy + by = passing through the origin. Corollary (2): If the straight lines are parallel, then h =ab [since 9 = 0°, tan9 = 0] Corollary (3): If the straight lines are perpendicular then coeff. of x 2 + coeff. of y 2 = [since 9 = 90°, tan9 = oo] 156 The condition for a general second degree equation 2 2 ax + 2hxy + by + 2gx + 2fy + c = to represent a pair of straight lines is abc + 2fgh - af - bg 2 -ch 2 = 2 2 Assume that ax + 2hxy + by + 2gx + 2fy + c = ■■■(!) represents a pair of straight lines. Treating this equation as a quadratic in x, ;an be written as ax + By solving for i,we get 2 2 this can be written as ax + 2(hy + g)x + (by + 2fy + c) = x = (hy + g)± yjjhy + R) 2 - a(by 2 + 2fy + c) ax ■ + hy + g = ± yj (hy + g) 2 - a(by 2 + 2fy + c) = ± \l(h 2 - ab)y 2 + 2(gh - af)y + (g 2 - ac) Now in order that each of these equations may be of the first degree in x and y, the expression in the R.H.S should be a perfect square. This is possible only if the discriminant of this quadratic in 'y' under the radical or within the root is zero. •'• (h - ab) (g - ac) = (gh - af) Simplifying this we get abc + 2fgh - af' - bg - ch =0 which is the required condition. 2 2 Example 5.28: Find the angle between the straight lines x + Axy + 3y =0 Solution: Here a = l,2h = 4,b = 3 If '9' is the angle between the given straight lines, then 9 = tan d 2W - ab a + 1 = tan 2^/4^3" = tan 2 2 Example 5.29: The slope of one of the straight lines of ax + 2hxy + by = is 2 thrice that of the other, show that 3h = Aab Solution: Let 'mj' and 'm 2 ' be the slopes of pair of straight lines. 2h a Then m, + m 2 = — ~r , m, m 2 = r It is given that m 2 = 3m j 2h m [ + 3m l = -~r => h_ 2b 157 2_a _^,fz*¥_ But m l . 3otj = t =^> 3m 1 = » ~^ D \2b ) 3h 2 a => 3/j 2 = 4a6 2 2 Example 5.30: Show that x - y +x-3y-2 = represents a pair of straight lines. Also find the angle between them. Solution: The given equation is x 2 -y 2 + x-3y-2 = ...(1) 2 2 Comparing this with ax + 2hxy + by + 2gx + 2fy + c = we get a = 1, 13 h = 0,b = -l,g = j,f=-j , c = - 2. Condition for the given equation to 2 2 2 represent a pair of straight lines is abc+2fgh-af -bg -ch = abc+2fgh-af-bg 2 -ch 2 = (l)(-l)(-2)+2(- 1) (£) (0)-(l) (f) - -1) (£)-(2) (0) 9 1 8-9+1 - 2 "4 +4 - 4 = Hence the given equation represents a pair of straight lines. Since a + &=l-l=0, the angle between the straight lines is 90°. 2 2 Example 5.31: Show that the equation 3x + Ixy + 2y + 5x + 5y + 2 = represents a pair of straight lines and also find the separate equation of the straight lines. Solution: 2 2 Comparing the given equation with ax + 2hxy + by + 2gx + 2fy + c = 0, we get 7 5 5 a = 3, b = 2, h = j , g = j , f = ^ , c = 2. The condition for the given equation to represent a pair of straight lines is abc + 2fgh - aj -bg - ch =0 abc+2fgh-aj 2 -bg 2 -ch 2 = (3) (2) (2) + 2(f) (f) (J) - 3 (f) -2 (f) -2^ 175 75 50 98 _ -I2+4 _ 4 _ 4 _ 4 _ 158 Hence the given equation represents a pair of straight lines. Now, factorising the second degree terms we get 3x 2 + Ixy + 2y 2 = (x + 2y) (3x + y) Let 3x 2 + Ixy + 2y 2 + 5x + 5y + 2 = (x + 2y + I) (3x + y + m) Comparing the coefficient of x, 31 + m = 5 ; Comparing the coefficient of y, 1 + 2m = 5 Solving these two equations, we get I = l,m = 2 :. The separate equations are x + 2y + 1 = and 3x + y + 2 = 2 2 Example 5.32: Show that the equation 4x + 4xy + y - 6x - 3y - 4 = represents a pair of parallel lines and find the distance between them. Solution: 2 2 The given equation is 4x + 4xy + y -6x-3y -4 = Here a = 4, h = 2, b=\ ; ab - h 2 = 4(1) -2 2 = 4-4 = .'. The given equation represents a pair of parallel straight lines. Now 4x 2 + 4xy + y 2 = (2x + y) 2 :. 4x 2 + 4xy + y 2 - 6x - 3y - 4 = (2x + y + I) (2x + y + m) Comparing the coefficient of x, 21 + 2m = - 6 i.e. l + m = -3 ■■■(!) Comparing the constant term, lm = -4 . . . (2) / + ^y-J =-3 ^ I 2 + 31-4 = i.e. (I + 4) (I - 1) = => I = - 4, 1 Now Im = -4 => to = 1, — 4 .". The separate equations are 2x + y - 4 = and 2x + y + 1 = I ci - C2 I -4-1 The distance between them is = ^/5 units yja 2 + b 2 \J2 2 + l 2 Example 5.33: Find the combined equation of the straight lines whose separate equations are x + 2y - 3 = and 3x - y + 4 = Solution: The combined equation of the given straight lines is (x + 2y-3)(3x-y + 4) = i.e. 3x 2 + 6xy - 9x - xy - 2y 2 + 3y + 4x + 8y - 12 = 2 2 i.e. 3x + 5xy — 2y - 5x + lly - 12 = is the required combined equation. 159 EXERCISE 5.4 2 2 (1) If the equation ax + 3xy - 2y - 5x + 5y + c = represents a pair of perpendicular straight lines, find a and c. (2) Find the angle between the pair of straight lines given by (a 2 - 3b 2 ) x 2 + Sab xy + fb 2 - 3a 2 )y 2 = (3) Show that if one of the angles between pair of straight lines ax 2 + 2hxy + by 2 =0 is 60° then (a + 3b) (3a + b) = Ah 2 (4) Show that 9x 2 + 2Axy + I6y 2 + 2lx + 2%y + 6 = represents a pair of parallel straight lines and find the distance between them. 2 2 (5) The slope of one of the straight lines ax + 2hxy + by = is twice that 9 of the other, show that 8h = 9ab. (6) Find the combined equation of the straight lines through the origin, one of which is parallel to and the other is perpendicular to the straight line 2x + y + 1 = (7) Find the combined equation of the straight lines whose separate equations are x + 2y - 3 = and 3x + y + 5 = 2 2 (8) Find k such that the equation 12* + Ixy - \2y -x + ly + k = represents a pair of straight lines. Find the separate equations of the straight lines and also the angle between them. 2 2 (9) If the equation 12x - lOxy + 2y + \Ax - 5y + c = represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them. 2 2 (10) For what value of k does 12* + Ixy + ky + I3x -y + 3 = represents a pair of straight lines? Also write the separate equations. (11) Show that 3x 2 + lOxy + 8y 2 + \Ax + 22y + 15 = represents a pair of _, (2^ straight lines and the angle between them is tan I tt 5.5 Circle Definition: A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle. 160 () 0P (x, y) Fig. 5.8 5.5.1 The equation of a circle when the centre and radius are given Let C(h, k) be the centre and r be the radius of the circle. Let P(x, y) be any point on the circle CP = r => CP 2 = r 2 => (x - h) 2 + (y-k) 2 = r 2 is the required equation of the circle. Note: If the centre of the circle is at the origin, i.e. (h, k) = (0, 0) then the equation of the 2 2 2 circle is x +y = r . 5.5.2 The equation of a circle if the end points of a diameter are given Let A(xj, Vj) and B(x 2 , y 2 ) be the end points of a diameter. Let P(x, y) be any point on the circle. The angle in a semi circle is a right angle. .". PA is perpendicular to PB .-. (Slope of PA) (Slope of PB) = - 1 fy-yi] (y-yi\ x o P (X, V) (A;. V 2 ) = -1 Fig. 5.9. \x - x\) \x — X2J (y -y{)(y- y 2 ) = -(*- *i) (x - x 2 ) :. (x - Xj) (x - x 2 ) + (y - Vj) (y - v 2 ) = is the required equation of the circle. 5.5.3 The general equation of the circle is x 2 + y 2 + 2gx + 2fy + c = 2 2 Consider the equation x + y + 2gx + 2fy + c= This can be written as x 2 + 2gx + g 2 + y 2 + 2fy +J 2 = g 2 +J 2 -c (x + g) 2 + (y +f) 2 = (^g 2 +f 2 - c ) 2 [x - (- g)f +[y- (-f)f = (V# 2 +/ 2 -c) 2 2 2 2 This is of the form (x-h) +(y-k) = r :. The considered equation represents a circle with centre (- g, —J) and radius ^g 2 +f 2 ■ 2 2 .'. The general equation of the circle is x +y + 2gx + 2fy + c = 161 2 2 Note : The general second degree equation ax +by + 2hxy + 2gx + 2fy + c = 2 2 represents a circle if (Y)a = b i.e. coefficient of x = coefficient of y (2) h = i.e. no xy term 5.5.4 Parametric form Consider a circle with radius r and centre at the origin. Let P(x, y) be any point on the circle. Assume that OP makes an angle 9 with the positive direction of x-axis. Draw the perpendicular PM to the x-axis. X V From the figure (5.10), ~ = cos9, ~; = sin9. Fig. 5.10 Here x and y are the co-ordinates of any point on the circle. Note that these two co-ordinates depend on 9. The value of r is fixed. The equations x = r cos9, y = r sin9 are called 2 2 2 the parametric equations of the circle x + y = r . Here '9' is called the parameter and < 9 < 2n Another parametric form: We know that sin 9 = 2 tany i 2~e 1 + tan 2 cos9 = 1 - tan z 2 i 2© 1 + tan 2 Let t = tan y If < 9 < 2n then - oo < t < oo x = r cos 9 ril - n x= ^~ i + r Thus x = rd - a l + t 2 2rt 2 2 2 equation of the circle x + y = r r(\ - t 2 ) 2rt y = r sin9 2rt y l + r co < f < co is another parametric Clearly x= -, ■ y = i + t 2 2 2 2 satisfy the equation x + y = r Example 5.34: Find the equation of the circle if the centre and radius are (2, - 3) and 4 respectively. 162 Solution: 2 2 2 The equation of the circle is (x-h) +(y-k) =r Here (h, k) = (2, - 3) and r = 4 .-. (x - 2) 2 + (y + 3) 2 = 4 2 2 2 i.e. x + y - 4x + 6y - 3 = is the required equation of the circle. Example 5.35: Find the equation of the circle if (2, - 3) and (3, 1) are the extremities of a diameter. Solution: The equation of the circle is (x - Xj) (x — x 2 ) + (y — y^) (y - y 2 ) = Here (x v y { ) = (2, - 3) and (x 2 , y 2 ) = (3, 1) .". (x-2)(x-3) + (y + 3)(y-l) = x 2 - 5x + 6 + y 2 + 2y - 3 = 2 2 .". The required equation isx +y -5x + 2y + 3 = 2 2 Example 5.36: Find the centre and radius of the circle x +>> +2x-4y + 3 = Solution: 2 2 The general equation of the circle is x + y + 2gx + Ify + c = Here 2g = 2, 2/= - 4, c = 3 .". centre is (- g, -/) = (— 1, 2) radius is *\jg +f-c= a/1 +4-3 = -\/2 units. 2 2 Example 5.37: Find the centre and radius of the circle 3x +3y -2x+6y -6 = Solution: 2 2 The given equation is 3x +3y - 2x + 6y - 6 = 2 2 ^ Rewriting the above, x +>> -tx + 2>>-2 = 2 2 Comparing this with the general equation x + y + 2gx + 2fy + c = We get 2g = -f,2/=2,c = -2 .-. centre is (-g,-f) = (j, - 1 radius is "\/g +f -c =a/q+1+2 = ^ units. Example 5.38: If (4, 1) is one extremity of a diameter of the circle 2 2 x + y - 2x + 6y - 15 = 0, find the other extremity. 163 Solution: Comparing x + y - 2x + 6y - 15 = with the general equation of the circle, A(4, I ) [ * ) B (*l < >i ) we get 2g = - 2 2f= 6 .-. centre is C (- g, -f) = (1, - 3) Fig. 5.11 Let B(x v yj) be the other extremity and A be (4, 1) C is the mid point of AB x\ +4 yi + 1 — 2~ = 1, ^— =-3 => x [= -2, y 1= -l :. The other extremity is (- 2, - 7) Example 5.39: Find the equation the circle passing through the points (0,1), (2,3) and (-2, 5). Solution: The general 2 equation of the circle is x + / '■ + 2gx + 2fy + c = The points (0, 1), (2, 3) and (- 2, 5) lie on the circle ••• 2/+c = - 1 ...(1) 4^ + 6/+ c = -13 ...(2) - 4g + 10/+ c = -29 ...(3) (l)-(2) => -4g-4f = 12 8+f = -3 ...(4) (2) - (3) => *g-¥ = 16 2g~f = 4 ...(5) (4) + (5) => 3g = 1 => g = i (4) => / = -3-1- 10 3 (1) => c = 17 3 • 2,2, .. x +y + 2@x + 2(- IG\ 17 :. 3x 2 + 3y 2 + 2jc - 20y + 17 = is the required equation. Example 5.40: Find the equation of the circle passing through the points (0, 1), (2, 3) and having the centre on the line x - 2y + 3 = 164 Solution: The general equation of the circle is =-1 ...(1) =-13 ...(2) :-g + 2f=-3 ...(3) -Ag-Af =12 i.e. g+f= -3 •••(4) 3/= -6 •••/=- 2 s = -i c = 3 x 2 + y 2 + 2gx + 2fy + c = (0, 1) lies on the circle .". 2f+ c (2, 3) lies on the circle .". Ag + 6f+ c The centre (- g, —J) lies on x - 2y + 3 = ; (1) - (2) => (3) + (4) => (3) => (1) => 2 2 .". The required equation is x +y -2x-4y + 3 = Example 5.41: Find the values of a and b if the equation 2 2 (a - 4)x + by + (b - 3)xy + Ax + Ay - 1 = represents a circle. Solution: 2 2 The given equation is (a - A)x + by + (b - 3)xy + Ax + Ay - 1 = (i) coefficient of xy = => b - 3 = .'. b = 3 2 2 t (ii) coefficient of x = co-efficient ofj => a- A = b :. a = 7 Thus a = 7, b = 3 Example 5.42: Find the equation of the circle with centre (2, - 3) and radius 3. Show that it passes through the point (2, 0). Solution: If the centre is (h, k) and radius is r, then the equation of the circle is (x-h) 2 + (y-k) 2 = r 2 . Here (h, k) = (2, - 3) and r = 3. (x - 2) 2 + (y + 3) 2 = 3 2 2 2 (x - 2) + (y + 3) = 9 is the required equation of the circle. Putting (2, 0) in the equation of the circle, we get L.H.S. = (2 - 2) 2 + (0 + 3) 2 = + 9 = 9 = R.Hi Hence the circle passes through (2, 0) 165 Example 5.43: Find the equation of the circle with centre (1, - 2) and passing through the point (4, 1) Solution: Let C be (1, -2) and P be (4, 1) 1 P(4, 1 ) Radius r = CP = -\j(l -4) 2 + (-2 - l) 2 = ^9 + 9 = a/T8 2 2 2 2 Thus the equation of the circle is (x - h) +(y-k) =r =r __ 2 Fig. 5.12 => (x-1) 2 + (v + 2) 2 = a/I8 2 2 i.e. x + >> -2x + 4>>-13=0 is the required equation. 2 2 Example 5.44: Find the parametric equations of the circle x + y =16 Solution: Here r = 16 => r = 4 . The parametric equations of the circle 2 2 2 x + y = r in parameter 9 are x = r cos9, y = r sin 9 2 2 .". The parametric equations of the given circle x + y = 16 are x = 4 cos 9, y = 4 sin 9, < 9 < 2n Example 5.45: Find the cartesian equation of the circle whose parametric equations are x = 2 cos 9, y = 2 sin 9, < 9 < 2% Solution: To find the caretsian equation of the circle, eliminate the parameter '9' x v from the given equations, cos 9 = x ; sin 9 = ~ cos 2 9 + sin 2 9 = 1 => ( |J + f |J =1 2 2 .". x + j = 4 is the required cartesian equation of the circle. EXERCISE 5.5 (1) Find the centre and radius of the following circles: (i) x 2 + y 2 = 1 (ii) x 2 + y 2 - 4x - 6y - 9 = (iii) x 2 + y 2 - 8x - 6y - 24 = (iv) 3x 2 + 3y 2 + 4x - Ay - 4 = (v)(jc-3)(jc-5) + (y-7)(y-l) = (2) For what values of a and b does the equation 2 2 (a - 2)x + by + (b — 2)xy + Ax + Ay - 1 = represents a circle? Write down the resulting equation of the circle. (3) Find the equation of the circle passing through the point (1, 2) and having its centre at (2, 3). 166 (6) (7) (8) (9) (4) x + 2y = 7, 2x + y = 8 are two diameters of a circle with radius 5 units. Find the equation of the circle. (5) The area of a circle is I6n square units. If the centre of the circle is (7,-3), find the equation of the circle. Find the equation of the circle whose centre is (- 4, 5) and circumference is 8ti units. 2 2 Find the circumference and area of the circle x +y -6x-8y+15 = Find the equation of the circle which passes through (2, 3) and whose centre is on x-axis and radius is 5 units. Find the equation of the circle described on the line joining the points (1,2) and (2, 4) as its diameter. (10) Find the equation of the circle passing through the points (1, 0), (0, - 1) and (0, 1). (11) Find the equation of the circle passing through the points (1, 1), (2, -1) and (3, 2). (12) Find the equation of the circle that passes through the points (4, 1) and (6, 5) and has its centre on the line 4x + y= 16. (13) Find the equation of the circle whose centre is on the line x = 2y and which passes through the points (- 1, 2) and (3, - 2). (14) Find the cartesian equation of the circle whose parametric equations are x = t cos9, y = t sin 9 and < 9 < 2n 2 2 (15) Find the parametric equation of the circle Ax +Ay =9 5.6. Tangent 5.6.1 Introduction Let us consider a circle with centre at C and a straight line AB. This straight line can be related to the circle in 3 different positions as shown in the following figures. (a) (b) (c) Fig. 5.13 167 In figure (5.13 a), the straight line AB does not touch or intersect the circle. In figure (5.13 b), the straight line AB intersects the circle in two points and it is called a secant. In figure (5.13 c), the straight line AB touches the circle at exactly one point, and it is called a tangent. In otherwords, the limiting form of a secant is called a tangent (Fig. 5.13d) Definition : A tangent to a circle is a straight line which intersects (touches) the circle in exactly one point. 5.6.2 Equation of the tangent to a circle at a point (x v y 1 ) Let the equation of the circle be x 2 + y 2 + 2gx + 2fy + c = ...(1) Let P(Xj, y j ) be a given point on it. .-. x 2 + y 2 + 2gx [ +2fy l +c = ...(2) Let PT be the tangent at P. The centre of the circle is C(— g, —J). Fig. 5.14 Slope of the CP = Since CP is perpendicular to PT, slope of PT = Equation of the tangent PT is yi+f xi +g x\+g yi +/, y-y l = w(x-Xj) x\ + 8 y-yi yi+f. (X - Xj) Cy - yO (yi+f) = -(x- *,) (x l + g) (y-yi)(y l +f) + (x-x l )(x l + g) = o => yy l -y 1 2 +fy-fy l + [xx l -x 2 + gx-gx l ] = o => xx l + yy l +fy + gx = x 2 + y 2 + gx { +fy { Add g*j +/y 1 + c on both sides xx l +yy l + gx + gx 1 +fy+fy l + c = x l +y l +2gx 1 +2fy l + c xx l + yy l + g(x + x±) + fly + y±) + c = is the required equation of the tangent at (x^, y^) 168 Corollary: 2 2 2 The equation of the tangent at (x { , y^) to the circle x + y = a is 2 xx, + yy, = a . 2 Note: To get the equation of the tangent at (Xj, y^), replace x as xxp 2 x + x\ y + y\ y as yy^, x as — j — anc ^ y as — ? — * n tne ec l uat i ori °f tne circle. 5.6.3 Length of the tangent to the circle from a point (x v y 1 ) Let the equation of the circle be x 2 + y 2 + 2gx + 2fy + c = Let PT be the tangent to the circle from [ it— — ^ ' P<xj, y ( ) P(xj, jj) outside it. We know that the co-ordinate of the centre C is (- g, —J) and radius r = CT = '\jg 2 +f 2 -c From the right angled triangle PCT, PT 2 = PC 2 - CT 2 = ( Xl+ g) 2 + (y 1+f) 2 -(g 2 + f 2 -c) = Xj +2gx l +g +y 1 +2fyj+f -g -f +c = x 1 +yj +2gx 1 +2fy 1 + c :. PT = a/xi + y\ + 2gx\ + 2fy\ + c , which is the length of the tangent from the point (x^,y^) 2 2 to the circle x + y + 2gx + 2fy + c = 2 Note : (1) If the point P is on the circle then PT = (PT is zero). 2 (2) If the point P is outside the circle then PT > (PT is real) 2 (3) If the point P is inside the circle then PT < (PT is imaginary) Corollary: The constant c will be positive if the origin is outside the circle, zero if it is on the circle and negative if it is inside the circle. 5.6.4 The condition for the line y = mx + c to be a tangent to the circle x 2 + y 2 = a 2 2 2 2 Let the line y = mx + c be a tangent to the circle x +y =a at (x\, y{) But the equation of the tangent at (x±, y^) to the circle 2 2 2- 9 x + y = a is xxy + yy^ = a 169 Thus the equations y = mx + c and xxj + yy l = a are representing the same straight line and hence their coefficients are proportional. 1 m c >'l x\ 2 - a m a 2 2 2 But (Xpjj) is a point on the circle x +y =a 2, 2 2 . . Xj +y, = a 2 2 2 2 am + a = c 4 2 4 a m a_ o + 1 = a a 2 (m 2 + 1) = c 2 i.e. 2 2 2 c =a (1 + m ) is the required condition. 2 2 2 Note:(l)The point of contact of the tangent y = mx + c to the circle x +y = a is - am a J\jl +m 2 ' -\jl +m 2 _ (2) The equation of any tangent to a circle is of the form y = mx ± a Ajl + m 5.6.5 Two tangents can be drawn from a point to a circle Let (x,, jj) be the given point. We know that y = mx±a\l + m is the equation of any tangent. It passes through (xj, y,). .". y, = mx, ±a\jl + m y< - mx, = + a \jl + m (y, -mx,) =a (1 + m ) a 2 m 2 = => y.+mx,- 2mx,y, - c => m (x, - a ) - 2mx,y~, + (y, - a ) = This is a quadratic equation in 'm'. Thus 'm' has two values. But 'm' is the slope of the tangent. Thus two tangents can be drawn from a point to a circle. Note : (1) If (Xp yj) is an exterior point (lies outside) then both the tangents are real and visible 170 Q(-*> v 2 ) ■>,) Fig. 5.16 (2) If (jcj, yj) is an interior point (lies inside) the circle then both the tangents are imaginary and hence not visible. (3) If (xp yj) is a boundary point (lies on) then both the tangents coincide and appears to be one. 5.6.6. Equation of the chord of contact of tangents from a point to the circle The general equation of the circle is x 2 + y 2 + 2gx + 2fy + c = ...(1) Let P(xj, >>j) be a point outside the circle. Let the tangents from P(xj, y^) touch the circle at Q(x 2 , y 2 ) and R(x 3 , y 3 ) The equation of the tangent PQ at Q (x 2 , y 2 ) * s xx 2 + yy 2 + g(x + x 2 ) +/(y + y 2 ) + c = The equation of the tangent PR at R(x 3 , y 3 ) is xx 3 + xy 3 + g(x + x 3 ) +/(y + j 3 ) + c = But (xj, yj) satisfy the equations (2) and (3) •'• x \ x l + >'l>'2 + ^^l + ^ + ^>'l + J2) + c = ° and XjX 3 + y x y 3 + g(x x + x 3 ) +/ijj + y 3 ) + c = But equations (4) and (5) show that (x 2 , y 2 ) and (x 3 , y 3 ) lie on the line xxj + yy 1 + g(x + Xj) +/(j + y t ) + c = Hence the straight line xxj + yy^ + g(x + Xj) +f(y + y^) + c = represents the equation of QR, chord of contact of tangents from (x J: y^). Example 5.46: Find the length of the tangent from (2, 3) to the circle x 2 + /-4x-3;y+12 = 0. Solution: 2 2 The length of the tangent to the circle x +y + 2gx + 2fy + c = from the (2) (3) (4) (5) point (x v yj) is y^ 2 + y 2 + 2gx x + 2fy x + c :. Length of the tangent to the given circle is ~\lx\ = V2 2 + 3 2 + yi 2 -4xi-3yi + 12 4.2-3.3+12 = ^4 + 9-8-9+ 12 = ^/8 = 2^/2 units 171 Example 5.47: Show that the point (2, 3) lies inside the circle x 2 + y 2 -6x-8y + 12 = 0. Solution: The length of the tangent PT from P(xj, jj) to the circle x + y + 2gx +2fy + c = is PT= -\jx l 2 + y l 2 + 2gx l + 2fy l + c PT 2 = 2 2 + 3 2 - 6.2 - 8.3 + 12 = 4 + 9 - 12 - 24 + 12 = - 11 < The point (2, 3) lies inside the circle 2 2 Example 5.48: Find the equation of the tangent to the circle x +y =25 at (4, 3). Solution: 2 2 The equation of the circle is x +y = 25 . The equation of the tangent at (xj, jj) is xxj + yy l = 25. Here (xj, y^ = (4, 3). .". The equation of the tangent at (4, 3) is 4x+3y = 25 2 2 Example 5.49: If >>=3x+c is a tangent to the circle x +y =9, find the value of c. Solution: The condition for the line y = mx + c to be a tangent to 2 2 2 / 2 x + y = a isc=±a'\jl+m Here a = 3, m = 3 .-. c = ± 3 VTO Example 5.50: Find the equation of the tangent to x 2 + y 2 - 4x + Ay - 8 = at (- 2, - 2) Solution: The equation of the tangent at (x\, y\) to the given circle is fx + x{\ fy + y{\ xx 1+}7l -4^-J +4(-2-J -8 = xxj + yy^ - 2 (x + Xj) + 2(y + y^) - 8 = At (- 2, - 2), the equation of the tangent is - 2x - 2y - 2 (x - 2) + 2(y - 2) - 8 = => - 4x - 8 = => x + 2 = is the required equation of the tangent. Example 5.51: Find the length of the chord intercepted by the circle 2 2 x +y -2x-y + I =0 and the line x - 2y = 1 . 172 Solution: To find the end points of the chord, solve the equations of the circle and the line. Substitute x = 2y + 1 in the equation of the circle. (2y + l) 2 + y 2 - 2(2y + 1) - y + 1 = Ay 2 + Ay + 1 + y 2 - Ay - 2 - y + 1 =0 5y 2 -y = y = x= 1 .•. The two end points are (1, 0) and [T ,T 7 1 y(5y - l) = l ? = 5 7 x =" .". Length of the chord = A / 1 1 s) H°-i) 2 T\ 2 25 + 25 : V5 units Example 5.52: Find the value of/? if the line 3x + Ay - p = is a tangent to the 2 2 circle x +>> =16. Solution: 2 2 2 The condition for the tangency isc = a (1 + m ) . 2 3d Here a =16, m = — t, c=^ c 2 = a 2 (l+m 2 ) l^= 16 l 1+ ^J =25 // = 16 x 25 :.p = ± 20 Example 5.53: Find the equation of the circle which has its centre at (2, 3) and touches the x-axis. Solution: Let P be a point on x-axis where it touches the circle. Given that the centre C is (2, 3) and P is (2, 0) Y •=CP = V(2-2) 2 + (3-0) 2 =3 2 2 2 The equation of the circle is (x - h) +(y-k) =r (x-2) 2 + (y-3) 2 = 3 2 o P(2. 0) -X Fig. 5.17 x 2 + y 2 - Ax - 6y + A = 173 (1 (2 (3 (4 (5 (6 (7 (8 (9 (10 (11 (12 (13 (14 EXERCISE 5.6 Find the length of the tangent from (1, 2) to the circle x + y 2 - 2x + Ay + 9 = x 2 + y 2 + 2x - A = and qual. 2 2 Find the equation of the tangent to the circle x +y -4x + 8y - 5 = at 2 2 Is the point (7, - 11) lie inside or outside the circle x +y - 10x = 0? the po 2 2 or inside the circle x +y - 5x + 2y - 5 = Prove that the tangents from (0, 5) to the circles x 2 + y 2 + 2x - A = and 2 2 x +y - )> + 1 = are equal. Find th (2, 1). Is the f Determine whether the points (- 2, 1), (0, 0) and (4, - 3) lie outside, on 2 2 or inside the circle x +y - 5x + 2y - 5 = Find the co-ordinates of the point of intersection of the line x + y = 2 2 2 with the circle x + y = 4 2 2 Find the equation of the tangent lines to the circle x + y = 9 which are parallel to 2x + y - 3 = Find the length of the chord intercepted by the circle x 2 + y 2 - 14jc + Ay + 28 = and the line x - ly + 4 = Find the equation of the circle which has its centre at (5, 6) and touches (i) x-axis (ii) j-axis 2 2 Find the equation of the tangent to x +y - 2x - lOy + 1 = at (- 3, 2) 2 2 Find the equation of the tangent to the circle x +y =16 which are (i) perpendicular and (ii) parallel to the line x + y = 8 2 2 Find the equation of the tangent to the circle x +y - 4x + 2y - 21 = at (1,4). Find the value of/? so that the line 3x + Ay -p = is a tangent to x 2 + y 2 - 64 = Find the co-ordinates of the middle point of the chord which the circle 2 2 x +y +2x + )'-3 = 0cuts off by the line y = x - 1 . 5.7. Family of circles Concentric circles: Two or more circles having the same centre are called concentric circles. Circles touching each other: Two circles may touch each other either internally or externally. Let C 1? C 2 be the centers of the circle and r^, r 2 be their radii and P the point of contact. 174 Case (1): The two circles touch externally. The distance between their centres is equal to the sum of their radii. (i.e.) CjC 2 = r j + r 2 Case (2) The two circles touch internally: The distance between their centres is equal to the difference of their radii. Cj C 2 - CjP Fig. 5.18 Fig. 5.19 Orthogonal circles: Definition: Two circles are said to be orthogonal if the tangent at their point of intersection are at right angles. Condition for two circles cut orthogonally Let the two circles be 2 2 x + y + 2gyX + 2/jj + Cj = and 2 2 x + y + 2g 2 x + 2f 2 y + c 2 = and cut each other orthogonally. Fig. 5.20 Let A and B be the centres of the two circles •'• A is (- g v - /j) and B is (- g 2 , - f 2 ) r x = -^g { 2 +f { 2 - c\ and r 2 = ^82 2 +f2 2 -C2 In the right angled triangle APB, AB 2 = AP 2 + PB 2 i.e. (-g l +g 2 ) 2 + (-f 1 +f 2 ) 2 = g l 2 + fl 2 -c 1 + g 2 2 +f 2 2 -c 2 => g\ +8 2 - 2 8i8 2 + fi + h - 2 /l/2 = 8[ +/i ~ c i + g2 + fl ~ c 2 => -2g i g 2 -2fJ 2 = -c l -c 2 i.e. 2g l g 2 + 2f l f 2 = c l + c 2 is the required condition for orthogonality. 2 2 Example 5.54: Show that the circles x +y -4x + 6y + 8 = and 2 2 x + y - 10x - 6y + 14 = touch each other. 175 2 2 x +y -4x -6y -9 = and passing through the point (- 4, - 5). Solution: The given circles are x 2 + y 2 -4x + 6y + 8 = ... (1) and x 2 + y 2 - lOx - 6y + 14 = ...(2) (1) => gl = - 2 /j = 3, Cj = 8. Centre is A(2, - 3) radius ^ = ^g\ 2 +f\ 2 -ci = yJ4 + 9 - 8 = a/5 (2) => g 2 = - 5,/ 2 = - 3, c 2 = 14. Centre is B(5, 3) radius r 2 = ^25 + 9-14 = a/20 = 2^/5 Distance between A and B = "\/(2 - 5) 2 + (- 3 - 3) 2 = V9 + 36 = ^45 =3a/5 = r l + r 2 :. The circles touch each other. Example 5.55: Find the equation of the circle, which is concentric with the circle x Solution: 2 2 The given circle isx +y -4x-6y -9 = Centre (- g, -f) is (2, 3) The circle passes through the point (- 4, - 5). .-. radius = "\/(2 + 4) 2 + (3 + 5) 2 = a/36 + 64 = ^100 = 10 2 2 2 The equation of the circle is (x-h) +(y-k) =r Here (h, k) = (2, 3), r = 10 .-. (jc - 2) 2 + (y - 3) 2 = 10 2 2 2 x + y - 4x - 6y - 87 = is the required equation of the circle. 2 2 Example 5.56: Prove that the circles x +y - 8x + 6y - 23 = and 2 2 x +>> - 2x - 5y + 16 = are orthogonal. Solution: The equations of the circle are x 2 + y 2 -8x + 6y-23 = ...(1) x 2 + y 2 -2x-5y+l6 = ...(2) 176 (1) => gl = -4, / 1 = 3, c 1 = -23 5 (2) => g 2 = - 1, / 2 = - 2 , c 2 = 16 Condition for orthogonality is 2g 1( g 2 + 2/j/ 2 =Cj + c 2 2g { g 2 + 2f x / 2 = 2(-4) (- 1) + 2 (3) (-§) - 8 - 15 = - 7 C j + c 2 = -23 + 16 = -7 2^2 + 2 /l/2 = c i +c 2 .". The two circles cut orthogonally and hence they are orthogonal circles. Example 5.57: Find the equation of the circle which passes through the point (1, 2) and 2 2 2 2 cuts orthogonally each of the circles x + y = 9 and x + y -2x + 8y -1 = Solution: 2 2 Let the required equation of the circle be x +y + 2gx + 2fy + c = •■•(!) The point (1,2) lies on the circle 1 + 4 + 2g + 4/+ c = 2g + 4f+c = -5 ...(2) 2 2 The circle (1) cuts the circle x +j = 9 orthogonally. 2«i« 2 + 2 /l/2 = c \ + c 2 => 2g(0)+2f(0) = c-9 :. c = 9 ...(3) Again the circle (1) cutsx 2 + y -2x + 8y - 7 = orthogonally. .-. 2g(-l) + 2/(4) = C -7 => -2g + 8f =9-7 = 2 => -S + 4/=l ...(4) (2) becomes 2g + 4/=-14 ••• g + 2/=-7 ...(5) (4) + (5) => 6/= -6 =>/= -1 (5) => g-2=-7 => g = -5 2 2 .". The required equation of the circle is x +y -10x-2y + 9 = 177 EXERCISE 5.7 (1) Show that the circles x + y -2x + 6y + 6 = 2 1 and* +y - 5x + by + 15 = touch each other. (2) Show that each of the circles x +y + 4y - 1 = 0, x +y +6x + y + 8=0 2 2 and x +y - Ax - Ay - 37 = touches the other two. (3) Find the equation of the circle concentric with the circle 2 2 x +y - 2x - 6y + A = and having radius 7. (4) Find the equation of the circle which is concentric with the circle 2 2 x + y - 8x + 12y + 15 = and passes through the point (5, 4) 2 2 (5) Show that the circle x + y - 8x - 6;y + 21 = is orthogonal to the circle x 2 + y 2 - 2y - 15 = (6) Find the circles which cuts orthogonally each of the following circles (i) x 2 + y 2 + 2x + Ay + 1 = 0, x 2 +y 2 - Ax+3 = and x 2 + y 2 + 6y + 5 = (ii) x 2 + y 2 + 2x + I7y + A = 0, x 2 + y 2 + Ix + 6y + 1 1 = and x 2 + y 2 -x + 22y + 3 = (7) Find the equation of the circle which passes through (1, - 1) and cuts 2 2 orthogonally each of the circles x +y + 5x-5y + 9 = and x 2 + y 2 - 2x + 3y - 1 = (8) Find the equation of the circle which passes through (1, 1) and cuts 2 2 orthogonally each of the circles x +y -&x-2y+ 16 = and x 2 + y 2 - Ax - Ay - 1 = 178 6. TRIGONOMETRY 6.1 Introduction: Trigonometry is one of the oldest branch of Mathematics. The word trigonometry means "triangle measurement". In olden days trigonometry was mainly used as a tool for use in astronomy. The early Babylonians divided the circle into 360 equal parts, giving us degrees, perhaps because they thought that there were 360 days in a year. The sine function was invented in India, perhaps around 300 to 400 A.D. By the end of ninth century, all six trigometric functions and identities relating them were known to the Arabs. In its earlier stages trigonometry was mainly concerned with establishing relations between the sides and angles of a triangle, but now it finds its application in various branches of science such as surveying, engineering, navigation etc. For every branch of higher Mathematics a knowledge of trigonometry is essential. 6.1.1 Angles: An angle is defined as the amount of rotation of a revolving line from the initial & \ positive angle position to the terminal position. \r- ft ( anti clockwise) Counter-clockwise rotations will be called positive and the clockwise will be /— negative angle called negative. / (clockwise) Consider a rotating ray OA with its a* y' end point at the origin O. p[„ <5 1 The rotating ray OA is often called the terminal side of the angle and the positive half of the x-axis (OX) is called the initial side. The positive angle 9 is |XOA (counter-clockwise rotation) The negative angle 9 is |XOA' (clockwise rotation) Note : 1 . one complete rotation (counter -clockwise) = 360° = 360 degree 2. If there is no rotation the measure of the angle is 0°. 6.1.2 Measurement of angles: ( i V h If a rotation from the initial position to the terminal position is I ttt: I of the revolution, the angle is said to have a measure of one degree and written as 1°. A degree is divided into minutes, and minute is divided into seconds. 179 i.e. 1 degree (1°) = 60 minutes (60') 1 minute (1') = 60 seconds (60") In theoretical work another system of measurement of angles is used which is known as circular measure. A radian is taken as the unit of measurement. 6.1.3 Radian measure: Definition: One radian, written as l 6 is the measure of an angle subtended at the centre O of a circle of radius r by an arc of length r. Note L. To express the measure of an angle as a real number, we use radian measure. 2. The word "radians" is optional and often omitted. Thus is given for a rotation, it is understood to be in radians. 3. V in l c indicates the circular measure. 6.1.4 Relation between Degrees and Radians Since a circle of radius r has a circumference of 2nr, a circle of radius 1 unit (which is referred to as an unit circle) has circumference 2it. When 9 is a complete rotation, P travels the circumference of an unit circle completely. Fig. 6. 3 If 9 is a complete rotation (counter-clockwise) then 9 = 2it radian. On the other hand we already know that one complete rotation (counter-clockwise) is 360°, consequently, 360° = 2 it radians or 180° = n radian. It follows that ,„ n , 180 ° 1° = yjyT radian and ■ 71 = 1 radian. Therefore 1° = 0.01746 radian (app.) and 7 1 radian = 180° x^ =57 16' (app.). Conversion for some special angles: degrees 30° 45° 60° 90° 180° 270° 360° Radians 6 % 4 71 3 n 2 % 3ic 2 2ic (Table 6.1) Example 6.1: Convert 3ic 1 (i) 150° into radians (ii) -7- into degrees (iii) t radians into degrees. 180 Solution: (i) 150° = n 5 150 x Toq radians = t n (ii) 3ji -7- radians = 3 ; x 18 °° = 135° 4 n (iii) t radians = 1 18 ° l i«n 7 4 X V = 4 x 180x 2l 6.1.5 Quadrants Let X'OX and YOY' be two lines at right angles to each other as in the fig. (6.4) we call X'OX and YOY' as x-axis and y-axis respectively. Clearly these axes divide the entire plane into four equal parts, called quadrants. 14° 19' 5" IV -*x "Y Fig. 6. 4 The parts XOY, YOX', X'OY' and Y'OX are known as the first, the second, the third and the fourth quadrant respectively. Angle in standard position: If the vertex of an angle is at O and its initial side lies along x-axis, then the angle is said to be in standard position. Angle in a Quadrant: An angle is said to be in a particular quadrant, if the terminal side of the angle in standard position lies in that quadrant. Example 6.2: Find the quadrants in which the terminal sides of the following angles lie. (ii) - 300° ©■ x«- 60° Y «) *\ Fig. 6. 5 a From Fig (6.5a) the terminal side of - 60° lies in IV quadrant. Y Fig. 6.5 b From Fig (6.5b) the terminal side of - 300° lies in I quadrant. v Fig. 6.5 c From Fig (6.5c) 1295° = 3x360°+180° + 35° The terminal side lies in III quadrant. 181 EXERCISE 6.1 (1) Convert the following degree measure into radian measure. (i) 30° (ii) 100° (iii) 200° (iv) - 320° (v) - 85° (vi) 7° 30' (2) Find the degree measure corresponding to the following radian measure (i,(f) (ii)^) (iii) -3 (iv) @f) (3) Determine the quadrants in which the following degrees lie. (i)380° (ii) -140° (iii) 1100° 6.2. Trigonometrical ratios and Identities 6.2.1 Trigonometrical ratios: In the co-ordinate plane, consider a point A on the positive side of x-axis. Let this point revolve about the origin in the anti clockwise direction through an angle 9 and reach the point P. Now |XOP = 9. Let the point P be (x, y). Draw PL perpendicular to OX. The triangle OLP is a right angled triangle, in which 9 is in standard position. Also, from the AOLP, we have OL = x = Adjacent side ; PL = y = opposite side ; OP = ^jx 2 + y 2 = Hypotenuse (= r > 0) The trigonometrical ratios (circular functions) are defined as follows : y The sine of the angle 9 is defined as the ratio ~. it is denoted by sin9. o e 3 N L A j-» 5i Fig. 6. 6 i.e. and sin 9 = cos 9 = tan 9 = 1 r x r 1 x cosecant value at 9 = ~~ = cosec 9 ; y ^ r secant value at 9 = ~ = sec 9 ; x *■ cotangent value at 9 = ~ = cot 9 ; y ^ Note : 1. From the definition, observe that tan 9 and sec 9 are not defined if x = 0, while cot 9 and cosec 9 are not defined if y = 0. 2. cosec 9, sec 9 and cot9 are the reciprocals of sin9, cos 9 and tan9 respectively. 182 Example 6.3: If (2, 3) is a point on the terminal side of 9, find all the six trigonometrical ratios. Solution: P(x, y) is (2, 3) and it lies in the first quadrant. .". x = 2, y = 3 ■ = ^x 2 + y 2 =^A + 9 = >/l3 x 2 C0Se = r=VT3 cosec 9 = 13 sec 9 Vl3 v 3 tan9=^=2 „ 2 cot 9 = t PR3) 3 -\ Fjg. 6. 7 Note : 1. From example (6.3), we see that all the trigonometrical ratios are positive when the terminal side of angle 9 lies in first quadrant. Now, let us observe the sign of trigonometrical ratios if the point on the terminal side of 9 lies in the other quadrants, (other than the first quadrant). Example 6.4: If (- 2, - 3) is a point on the terminal side of 9. Find all the six trigonometrical ratios. Solution: P(x, y) is (- 2, - 3) and it lies in the third o quadrant / ••• x =- 2 , y = - 3 ; ( _2,- 3 ) r = \jx 2 + y 2 =A /4T9 =-y/l3 Fig. 6. 8 TV sin 9 = T r = .1. = = -ve;cose = - r = ^= = -ve;tmQ=^ = - yi3 3 ' ■_3 -2 = 2 = + ve cosec 9 = ~ = -*- 1 r~ = - ve ; sec 9 = ~ = r x 13 A -2 — =-ve ; cot 9 = ~r = 3 = + ve As example illustrates, trigonometric functions may be negative. For y r instance, since r is always positive, sin 9 = and cosec 9 = ~ have the same sign as y. Thus sin 9 and cosec 9 are positive when 9 is in the first or second quadrants, and negative when 9 is in the third or fourth quadrants. The signs of the other trigonometric functions can be found similarly. The following table indicates the signs depending on where 9 lies. 183 Quadrants Functions I II Ill IV Sin + + - - Cos + - - + Tan + - + - Cosec + + - - Sec + - - + Cot + - + - II sin cosec I All III tan cot IV cos sec Table (6.2) 6.2.2 Trigonometrical ratios of particular angles: Let X'OX and YOY' be the co-ordinate axes. With O as centre and unit radius draw a circle cutting the co-ordinate axes at A, B, A' and B' as shown in the figure. Suppose that a moving point starts from A and move along the circumference of the circle. Let it cover an arc length. 9 and take the final position P. Let the co-ordinates of this point be P(x, y). Then by definition, x = cos9 and y = sin9. We consider the arc length 9 to be positive or negative according as the variable point moves in the anti clockwise or clockwise direction respectively. Range of cos 9 and sin9 : Since for every point (x, y) on the unit circle, we have - 1 < x < 1 and - 1 < y < 1, therefore - 1 < cos 9 < 1 and - 1 < sin9 < 1 n 3n Values of cos9 and sin9 for 9 = 0, ~ » n, ~y and 2ti. We know that the circumference of a circle of unit radius is 2rc. If the moving point starts from A and moves in the anti clockwise direction then at the points A, B, A', B' and A the arc lengths covered are 9 = 0, y , n, ~j~ and 2% respectively. Also, the co-ordinates of these points are:A(l, 0), B(0, 1), A'(— 1, 0), B'(0,-l)andA(l,0) At the point: A(l, 0), 9 = => cosO =1 and sin = 184 B (0, 1), = f => K COST = and sin T =1 A' (-1, 0), Q = n => cos n = -1 and sin n = B' (0, - 1), 9-3f => cos 3 y = and sin 3 x = A (1,0), Q = 2n => cos 2tc = 1 and sin 2n = 6.2.3 Trigonometrical ratios of 30°, 45° and 60° Consider an isosceles right-angled triangle whose equal sides are 1. Its hypotenuse is ^/l 2 + l 2 = ^2 .Its base angle is 45°. .-. sin 45° = -;= ; cos 45° = -7= ; tan45° = 1 cosec 45° = ^2 ; sec 45° = ^2 ; cot 45° = 1 Fjg. 6. 10 Opposite side = 1 adjacent side = 1 hypotenuse = yJ2 Consider an equilateral triangle ABC of side 2 units. Each of its angle is 60°. Let CD be the bisector of angle C. Then angle ACD is 30°. Also AD = 1 ndCD = V2 2 -l 2 = V3 . Now in the right ar igled triangle ACD For 30° For 60° opposite side = 1 opposite side = y]3 adjacent side = -\J3 C adjacent side = 1 hypotenuse = 2 hypotenuse = 2 sin 30° = 2 2 /3(f A sin 60° = \ ™n "\/3 £<& , cos 30° = 2 A 1 D 1 B cos 60° = 2 tan 30°=^ Fig. 6.11 tan 60° = a/3 .'. cosec 30° = 2 .". cosec 60° = —fc ••• sec 30 ° = J3 :. cot 30° = ^3 sec 60° = 2 , cot 60° = ^ 185 e K n 7t 71 n 371 2tc 6 4 3 2 2 sin9 1 2 1 V2 2 1 -1 cos 9 1 2 1 1 2 -1 1 tan 9 1 V3 1 V3 CO — 00 Table 6.3 Important results: For all values of 9, cos (- 9) = cos 9 and sin (- 9) = - sin 9 Proof: Let X'OX and YOY' be the co-ordinate axes. With O as centre and unit radius draw a circle meeting OX at A. Now let a moving point start from A and move in anti clockwise direction and take the final position P(x, y) so that arc AP = 9. On the other hand, if the point starts from A and moves in the clockwise direction through the arc length AP' equal to arc length AP. Then arc AP'= - 9. Pfe-jj) Fig. 6. 12 Thus | AOP = 9 and | AOP' =-9 From the co-ordinate geometry, we know that the co-ordinates of P' are (x,-y). Clearly, cos9 and cos(-9) are respectively the distances of points P and P' from y axis and clearly each one of them is equal to x. :. cos (- 9) = cos 9 Clearly, sin 9 and sin(- 9) are respectively the distances of points P and P' from x-axis. As sin9 = y and sin (-9) = — y, we have sin(-9) = - sin 9 Deductions cosec (-9) = - cosec 9 ; sec (-9) = sec 9 tan (-9) = - tan 9 ; cot (-9) = - cot 9 186 6.2.4 T-ratios of (90° ± 9), (180° ± 9), (270° ± 9) and (360° ± 9): It is evident that, when 9 is a small angle (0 < 9 < 90°), then 90° - 9, 90° + 9 etc., are in the quadrants as given below: Angle Quadrant 90° -0 Ql (first quadrant) 90° + 9 Q2 180° - Q 2 180° + Q 3 270° - Q3 270° + 9 Q 4 360° - 9 ; also equal to "- 9" Q 4 360° + Qi Table 6.4: Let P(a, (3) be a point in the first quadrant. Let |XOP = 9 /. sin 9 = 7777 ; cos9 = 7777 ; tan 9 = OP a OP n 0P n 0P ^ a cosec 9 = ~tt ; sec 9 = — ; cot9 = 77 (3 a (3 T-ratios of (90° -9) Let Q be a point in the first quadrant such that Fig. 6. 13 |XOQ = 90° - and OQ = OP. Let PA and QB be perpendicular to OX and OY respectively. Then AOAP = AOBQ and Q is (p\ a). Hence sin (90° - 0) = cos (90° - 9) = y co-ordinate of Q a OQ OP x co-ordinate of Q (3 OQ OP = cos 9 = sin 9 , nnn ., y co-ordinate of Q a tan < 90 -V=x coordinate of Q = p = cot9 Similarly, cosec (90° - 9) = sec 9 sec (90° - 0) = cosec 9 cot (90° - 0) = tan 9 187 T-ratiosof(90° + 9) Let R be a point in the second quadrant such that |XOR =90° + 9andOR = OP Let RC be perpendicular to x axis. Then AOAP = ARCO and R is (- (3, a), Hence y co-ordinate of R sin (90° + 9) = l q^ : Y R(-M e\ P C O a t v x cos (90° + 9) = x co-ordinate of R OR y co-ordinate of R tan ( 90 ° + 9 ) - .coordinate of R Similarly, cosec (90° + 9) = sec 9 sec (90° + 9) = - cosec 9 cot (90° + 9) = - tan 9 T-ratiosof(180°-9) Let S be a point in the second quadrant such that |XOS =180° -9 and OS =OP Draw SD perpendicular to x-axis Thus A OAP = A ODS and S is (- a, (3). Hence Fig. 6. 14 a a Op" = cos9 : ^p = - sin 9 a : ^jt = - cot 9 Y S(- «P) A P(«,P) V Xe D O A X sin (180° cos (180° tan (180° y co-ordinate of S 9) = QS 9) = x co-ordinate of S OS y co-ordinate of S ' ~ x coordinate of S OP - a OP A. - a Fig. 6. 15 = sin 9 = - cos 9 = -tan 9 Similarly, cosec (180° - 9) = cosec 9 sec (180° -9) = -sec 9 cot (180° -9) = -cot 9 T-ratiosof(180° + 9) Let T be a point in the third quadrant such that |XOT = 180° + 9 and OT = OP Draw TE perpendicular to x-axis Then AOAP = AOET and T is (- a, - (3). Hence P(aP; 188 sin (180° + 9) = cos (180° + 9) = tan (180° + 9) = y co-ordinate of T -(3 OT = OP x co-ordinate of T - a OT = OP y co-ordinate of T -(3 = - sin 9 = - cos 9 = tan 9 Similarly, cosec (180° + 9) = sec (180° + 9) = x coordinate of T - a - cosec 9 - sec 9 cot (180° + 9) = cot 9 Remark: To determine the trigonometric ratios of any angle, follow the procedure given below n (i) Write the angle in the form & x + 9 ; k e Z. (ii) Determine the quadrant in which the terminal side of the angle lies, (iii) Detrmine the sign of the given trigonometric function in that , • S A , particular quadrant, using ™ ^ rule. (iv) If k is even, trigonometric function of allied angle equals the same function of 9. (v) If k is odd, then adopt the following changes: sine <-> cos ; tan <->■ cot ; sec <-> cosec Trigonometrical ratios for related angles Xngle function" -e 90 -e 90 + 9 180-9 180+9 270-6 270+9 360-9 or -9 sin -sin9 cos 9 cos 9 sin 6 - sin 9 -cos e -cos 9 -sin6 cos cos 8 sin 9 -sin 9 -cos 9 -cos 9 -sine sin 9 cos 9 tan -tanG cot 9 -cote -tan 9 tan 9 cote -cot 9 -tan9 cosec -cosecG sec 9 sec 9 cosec 9 -cosec9 -sec 6 -sec 9 -cosec9 sec sec 9 cosec 9 -cosec9 -sec 9 -sec 9 -cosece cosec 9 sec 9 cot -cote tan 9 -tan9 -cote cote tan6 -tan 9 -cot 9 Table 6.5 Note : Since 360° corresponds to one full revolution, sine of the angles 360 o +45 o ;720 o +45 o ;1080 o +45 o are equal to sine of 45°. This is so for the other trigonemetrical ratios. That is, when an angle exceeds 360°, it can be reduced to an angle between 0° and 360° by wiping out integral multiples of 360°. 189 Example 6.5: Simplify : (i) tan 735° (ii) cos 980° (iii) sin 2460° (iv) cos (-870°) (v) sin (-780°) (vi) cot (-855°) (vii) cosec 2040°(viii)sec (- 1305°) Solution: (i) tan (735°) = tan (2 x 360° + 15°) = tan 15° (ii) cos 980° = cos (2 x 360° + 260°) = cos 260° = cos (270° - 10°) = - sin 10° (iii) sin (2460°) = sin (6 x 360° + 300°) = sin (300°) = sin (360° - 60°) = - sin 60° _ 2 (iv) cos (- 870°) = cos (870°) = cos (2 x 360° + 150°) = cos 150 = cos (180° - 30°) = -cos 30° = -\- (v) sin (- 780°) = - sin 780° = - sin (2 x 360° + 60°) = - sin 60° = - *2 (vi) cot (- 855°) = - cot (855°) = - cot (2 x 360° + 135°) = - cot (135°) = - cot (180° - 45°) = cot 45° = 1 (vii) cosec (2040°) = cosec (5 x 360° + 240°) = cosec (240°) = cosec (180° + 60°) = - cosec (60°) = "^ (viii) sec (- 1305°) = sec (1305°) = sec (3 x 360° + 225°) = sec (225°) = sec (270° - 45°) = - cosec 45° = - ^2 ,,,„.,.* cot (90° -9) sin (180° + 9) sec (360° - 9) Example 6.6: Simplify : — — - e) - - e) cqs (9QO - Q) 190 . tan 9 (- sin 9) (sec 9) Solution: The given expression = — zrrr — : — rr tan 9 (sec 9) (- sin 9) = 1 Example 6. 7: Without using the tables, prove that sin 789° sin 489° + cos 129° cos69° = ~ Solution: sin 780° = sin (2 x 369° + 69°) = sin 69° = ^ sin 489° = sin (369° + 129°) = sin 129° = sin (189° - 69°) = sin 69° = \ cos 129° = cos (189° - 69°) = - cos 69° = - j \ cos69° = ^ L.H.S. - 2 -2 2 2 \/3 \/3 I I 2 • 2 = 4 "4 =2 RHS - 6.2.5 Special properties of Trigonometrical functions: Periodic function: A function fix) is said to be a periodic function with period a if fix + a)=fix). The least positive value of a is called the fundamental period of the function. All the circular functions (trigonometrical functions) are periodic functions. For example, sin (x + 2n) = sin x ; sin (x + 4%) = sin x ; sin (x + 6n) = sin x sin (x + 2nn) = sin x, n e Z Here a = - 6n, - 4ti, - 2n, 0, 2n, 4n, ... .But the fundamental period must be the least positive quantity. Therefore a = 2n is the fundamental period. Thus sine function is a periodic function with fundamental period 2n. Similarly one can prove that the functions cos x, cosec x and sec x are also periodic functions with fundamental period 2n while tan x and cot x are periodic with fundamental period n. 191 6.2.6 Odd and even functions: We know that, if fix) =fi- x), then the function is an even function and if fi- x) = -fix) then the function is an odd function. Consider fix) = sin x ; fi- x) = sin (-%) = - sin x = -fix) i.e. fix) = -fi- x) .". sin x is an odd function. Similarly we can prove that cosec x, tan x and cot x are odd functions. Consider fix) = cos x ;/(- x) = cos (- x) = cos x =fix). .'. cosx is an even function. Similarly we can prove sec x is an even function. Note : We can read more about odd and even function in Chapter 7. EXERCISE 6.2 (1) If sin 9 = y2 , find the value of sec (360° - 9) . tan (180° - 9) + cot (90° + 9) sin (270° + 9) (2) Express the following as functions of positive acute angles: - (i) sin (- 840°) (ii) cos (1220°) (iii) cot (- 640°) (iv) tan (300°) (v) cosec (420°) (vi) sin (- 1110°) (vii) cos (- 1050°) ,„, „ , sin 300° . tan 330° . sec 420° fl (3) Prove that — ^ . cos 210 ° . cosec 315° = ~ V3 71 (4) Prove that i 1+cot a - sec I a + ~ I ( 1 1 + cot a + sec I a + ~ I ( =2 cot a (5) Express the following as functions of A : (i)sec[A-^"J (ii) cosec [A -^ J (iii) tan I A - -y (iv) cos (720° + A) (v) tan (A + ti) ,, N „ , sin (180°+A) . cos (90°-A) . tan (270°-A) . A lk (6) Prove that , C/irio — tt ,,,.. .. ,,„ no .. = -sin A cos A sec (540° -A) cos (360°+A) cosec (270°+A) (7) Prove that sin 9 . cos 9 i sin I y - 9 J . cosec 9 + cos I y - 9 J sec 9 r = 1 (8) Find the values of :- (i) cos (135°) (ii) sin (240°) (iii) sec (225°) (iv) cos (- 150°) 5tc ( 5ti (v) cot (315°) (vi) cosec (- 300°) (vii) cot -j- (viii) tan (9) If A, B, C, D are angles of a cyclic quadrilateral prove that cosA + cosB + cos C + cos D = 0. 192 (10) Find the values of the following expressions : (i) tan 2 30° + tan 2 45° + tan 2 60° (ii) sin t . cos j + cos t . sin t II Jl Jl Jl (iii) cost, cost -sinT.sinT (iv) cos 45°.cos60°-sin 45°. sin 60° (v) tan 2 60° + 2 tan 2 45° (vi) tan 2 45° + 4 cos 2 60° (vii) cot 60° . tan30° + sec z 45° . sin 90° (viii) tan 2 60° + 4 cot 2 45° + 3 sec 2 30° + cos 2 90° (ix) tan 2 30° + 2 sin 60° + tan 45° - tan 60° + cos 2 30° (x) 2 sin 2 60° - j sec 60° tan 2 30° + 5 sin 2 45° . tan 2 60° (11) If cos 9 = - o and tan 9 > show that —f= ~ — - — : — r = 3. z -y3 cos 9 - 3 sin 9 6.2.7 Trigonometrical identities: 2 2 As in variables, sin 9 . sin 9 = (sin 9) . This will be written as sin 9. Similarly 2 ^ tan 9 tan 9 = tan 9 etc. We can derive some fundamental trigonometric identities as follows: Consider the unit circle with centre at the origin O. Let P(x, y) be any point on the circle with |XOP = 9. Draw PL perpendicular to OX. Now, triangle OLP is a right angled triangle in which (hypotenuse) OP = r = 1 unit, and x and y are adjacent and opposite sides respectively. x y y Now we have cos 9 = j = x and sin 9 = 7" = y; tan 9 = ~_ and r 2 = x 2 + y 2 = 1 2 2 2 From AOLP, we have x +y = r =1 Fig. 6.17 i.e. x 2 + y 2 = cos 2 9 + sin 2 9 = 1 1 + tan z 9 2._ 1+ £.£t£.(iy_(_L_p.._j x x ..2 .2 , ..2 Vcos 9 sec z 9 1 + COt2e = 1+X ~2 JL f L = g) 2 = {^tof = -sec 2 9 Thus we have the identities 193 sin 2 9 + cos 2 9 = 1 1 + tan 2 9 = sec 2 9 1 + cot 2 9 = cosec 2 9 From these we also have sec 2 9 - tan 2 = 1 2 2 cosec 9 - cot 9=1 Example 6.8: Show that cos A - sin A = 1 - 2 sin A Solution: cos 4 A - sin 4 A = (cos 2 A + sin 2 A) (cos 2 A - sin 2 A) = cos 2 A - sin 2 A = 1 - sin 2 A - sin 2 A = 1 - 2sin z A 2 2 2 2 Example 6.9: Prove that sec A + cosec A = sec A . cosec A Solution: 2 2 ' sec A + cosec A = 2 2 cos A sin A sin A + cos A ~~ 2* ■ 2* _ 2* • 2* cos A . sin A cos A . sin A 2 2 = sec A . cosec A Example 6.10: Show that cosA "\jl + cot 2 A = ^/cosec 2 A - 1 J Solution: cosA "\j 1 + cot A = cos A ^/cosec A = cosA . cosecA C0S A . / 2l 7 = ^^ = cotA = ^cosec A - 1 9 o j C — b Example 6.11: If a sin 9 + b cos 9 = c, show that tan 9 = ' a — c Solution: 2 2 a sin 9 + b cos 9 = c. 2 2 2 Dividing both sides by cos 9, we get a tan 9 + b = c sec 9 a tan 2 9 + b = c (1 + tan 2 9) tan Q (a - c) = c - b 2n c ~b :. tan 9 = Example 6.12: that 1 - cosA 1 + cosA a — c = cosec A - cotA _ . . 1 - cosA 1 - cosA 1 - cosA Solution: consider, i + CO sA = 1 + cosA X T^A 194 (1-cosA) 2 (\ - cosAV l-cos 2 A V sinA 1 - cosA 1 - cosA 1 cosA 1 + cosA ~~ sinA ~~ sinA sinA = cosec A - cotA Example 6.13: 2 2 2 2 If x = a cos9 + b sin 9 and y = a sin 9 - b cos 9, show that x +y = a + b 2 2 2 2 Solution: x + y = (a cos 9+ b sin9) + (a sin9 - b cos 9) 2 2 2 2 = a cos 9 + b sin 9 + lab cos 9 sin9 2 2 2 2 + a sin 9 + b cos 9 - lab sin9 cos9 = a 2 (cos 2 9 + sin 2 9) + 6 2 (sin 2 9 + cos 2 9) = a 2 + b 2 2 2 Example 6.14:Show that sin A.tanA+cos A . cotA+2 sinA . cosA=tanA + cotA Solution: L.H.S. = sin A . » + cos A . • » + 2sinA cosA 3 3 sin A cos A _ . = ^A +^A~ +2smA - cosA _ sin 4 A + cos 4 A + 2sin 2 A . cos 2 A sinA . cosA (sin 2 A + cos 2 A) 2 1 sinA . cosA ~~ sinA . cosA 2 2 sin A + cos A r . o . o . , n = sinA.cosA [vsm 2 A + cos 2 A=l] 2 2 sin A cos A sinA cosA sinA cosA citi A cosA Hence the result . + ■ . = tanA + cotA = R.H.S. Example 6.15: Show that 3(sinx-cosx) +6(sinx + cosx) +4 (sin x+cos x) = 13 2 4 r 2~l 2 2 2 Solution: (sinx - cosx) = [(sinx - cosx) J = [sin x + cos x - 2sinx cosx] 2 = [1 -2sinxcosx] 2 2 = 1 - 4sinx cosx + 4sin x cos x (i) 2 2 2 (sinx + cosx) = sin x + cos x+ 2sinx . cosx = 1 + 2sinx cosx (ii) f\ f\ 9 % % sin x + cos x = (sin x) + (cos x) 195 2 2 3 2 2 2 2 = (sin x + cos x) - 3sin * . cos x (sin x + cos x) 2 2 = 1 - 3sin x cos x (iii) Using (i), (ii) and (iii)L.H.S. 2 2 = 3(1 - 4sin x cos* + 4sin x . cos x) 2 2 + 6(1 + 2sinx cosx) + 4(1 - 3 sin x cos x) = 3+6+4 = 13 = R.H.S. , , „ , _ , tan9 + sec9 - 1 1 + sin9 Example 6.16: Prove that — — r = 7~ tan9 - sec9 + 1 cos9 Solution: L.H.S. = tan9 + sec9 - (sec 2 9 - tan 2 9) tan9 - sec9 + 1 tan9 + sec9 - (sec9 + tan9) (sec9 - tan9) tan9 - sec9 + 1 (tan9 + sec9) (1 - sec9 + tan9) = — — — = tan9 + sec9 (tan9 - sec9 + 1) sin9 1 _ sin9 +1 _ cos9 cos9 ~~ cos9 EXERCISE 6.3 (1) Prove the following: (i) sin A - cos A = 1 - 2cos A (ii) sin 3 A - cos 3 A = (sinA - cos A) (1 + sinA cosA) (iii) (sin9 + cos9) 2 + (sin9 - cos9) 2 = 2 (iv) (tan9 + cot9) = sec 9 + cosec 9 , - 1 1 2^ , -n secx + tanx , N ? (v) ', r~r + r~z = 2sec 9 (vi) = (secx + tanx) 1 + sin9 1 - sin9 secx - tanx (ix) cosec 9 „ 1 — — r = cos9 + tan9 1 1 + cos9 (vii) — TK — I — 7: = cos9 (viii) : — „ r = sec9 - tan9 v ' cot9 + tan9 tan9 + sec9 cosec9 - cot9 sin9 (x) (sec9 + cos9) (sec9 - cos9) = tan 9 + sin 9 (2) If tan9 + sec9 = x, show that 2tan9 = x - — , 2sec9 = x + — Hence show that sin 9 = ~ x 2 -l x"+l 196 2 2 l — (3) If tan9 + sin6 = p, tan8 - sin6 = q and p > q then show that p -q = 4^jpq sccA cosccA (4) Prove that (1 + cotA + tanA) (sinA - cos A) = y~ 5 — cosec A sec A ,_ _ , cosA sinA . (5) Prove that - — ; — r + — = sin A + cos A ' 1 - tanA 1 - cotA (6) Prove the following : / 1 + sinA A A ,_ /1+cosA W "\ / 1 -""a" = sec ^ + tatL ^ ( u ) ^ ' '-• ~ = cosec A + cotA 1 - sinA 1 - cosA (sinA * 1) (cosA * 1) /1-sine (111) \ I \ . . ~ = sec9 - tan9 1 + sin9 (7) If cos9 + sin9 = -\[2 cos9, show that cos9 - sin9 = -y2 sin9 (8) Prove that (1 + tanA + sec A) (1 + cotA - cosec A) = 2 6.3 Compound Angles 6.3.1 Compound Angles A + B and A - B In the previous chapter we have found the trigonometrical ratios of angles such as 90°+ 9, 180° + 9, ... which involves single angle only. In this chapter we shall express the trigonometrical ratios of compound angles such as A + B, A - B, ... interms of trigonometrical ratios of A, B, .... It is important to note that the relation/(x + y) =f(x) +f(y) is not true for all functions of a real variable. As an example all the six trigonometrical ratios do not satisfy the above relation. cos (A + B) is not equal to cosA + cosB. Let us develop the identity cos(A - B) = cosA cosB + sinA sinB ftp* Q(ix*iB,anB) Fig. 6. 18 Fig. 6. 19 197 Let P and Q be any two points on the unit circle such that |XOP = A and |XOQ = B. Then the coordinates of P and Q are (cos A, sinA) and (cosB, sinB) respectively. PQ 2 = (cosA - cosB) 2 + (sinA - sinB) 2 = (cos A - 2cosA cosB + cos B) + (sin A - 2sinA sinB + sin B) = (cos 2 A + sin 2 A) + (cos 2 B + sin 2 B) - 2cosA cosB - 2sinAcosB = 1+1 - 2 cos A cosB - 2 sinA sinB=2 - 2 (cosA cosB+sinA sinB) ... (1) Now imagine that the unit circle above is rotated so that the point Q is at (1, 0). The length PQ has not changed. PQ 2 = [cos (A - B) - l] 2 + [sin (A - B) - 0] 2 = [cos 2 (A - B) - 2cos(A - B) + l] + sin 2 (A - B) = [cos 2 (A - B) + sin 2 (A - B)] + 1 - 2cos (A -B) = 1 + 1 - 2cos(A - B) = 2-2cos(A-B) ...(2) From (1) and (2), 2 - 2cos(A - B) = 2 - 2 (cosAcosB + sinA sinB) => cos(A - B) = cosA cosB + sinA sinB Next let us consider cos(A + B). This is equal to cos [A - (- B)] and by cosine of a difference identity, we have the following: cos(A + B) = cosA cos(- B) + sinA . sin (- B) But cos(- B) = cosB and sin(-B) = - sin B .'. cos(A + B) = cosA cosB - sinA sinB. To develop an identity for sin(A + B), we recall the following: (n sin9 = cos I 2~ 9 In this identity we shall substitute A + B for sin (A + B) = cos f-(A + B) = cos f-Al-B We can now use the identity for the cosine of a difference. = cosl2"-Aj . cosB + sin I 2~ A | .sinB = sinA . cosB + cosA . sinB Thus, sin (A + B) = sinA . cosB + cosA sinB 198 To find an identity for the sine of a difference, we can use the identity just derived, substituting - B for B sin (A - B) = sin [A + (- B)] = sinA cos(- B) + cosA . sin(- B) sin(A - B) = sinA cosB - cosA sinB An identity for the tangent of a sum can be derived using identities already established. , A _ sin (A + B) tan(A + B) = cos(A + B) sinA . cosB + cosA sinB cosA cosB - sinA sinB Divide both Numerator and Denominator by cosA cosB sinA cosB cosA sinB cosA cosB cosA cosB - cosA cosB sinA sinB cosA cosB cosA cosB tanA + tanB tan (A + B) = - — ; — : — : — — v ' 1 - tanA . tanB Similarly, an identity for the tangent of a difference can be established. T ■ ■ , , . t^ tan A - tan B It is given by tan (A - B) = - — : — : — : — ~ b J ■ ' 1 + tanA . tan B (1) sin (A + B) = sinA cosB + cos A sinB (2) sin (A - B) = sinA cosB - cosA sinB (3) cos (A + B) = cosA cosB - sinA sinB (4) cos (A - B) = cosA cosB + sinA sinB tanA + tanB (5) tan (A + B) = 1 - tanA tanB tanA - tanB (6) tan ( A - B )=l + tanA.tanB Example 6.i7:Find the values of (i)cos 15° (ii)cos 105° (iii)sin 75° (iv)tan 15° Solution: (i) cosl5° = cos(45°-30°) = cos45° cos30° + sin45° sin30° _J_ i/l 1 1 a/3+1 a/3 + 1 a/2 \/6+\/2 "f2 2 + ^2- 2 ^ - 2f2 ^ " 4 (ii) cosl05° = cos(60° + 45°) = cos60° cos45° - sin60° sin45° 199 _I J_ _i/l J_ _ l->/3 a/2-\/6 -2^/2 2^-2^- 4 (iii) sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° _J_ i/1 J_ 1 a/3+1 a/6 + a/2 ->/2 • 2 + ^2 -2- 2a /2 - 4 „,„ ™- tan45° - tan30° \/3 (iv) tanl5° = tan(45° - 30°) = tan450 = — — p- 1+1 -Vf 3-\/3 = 2-a/3 3+a/3 A, B are Solution: cos (A + B) = cosA cosB - sinA sinB 3 12 Example 6.18: If A, B are acute angles, sinA = t ;cos B= tt , find cos(A + B) H- cosA = yj 1 - sin A = -\ 4 - 5 X/ 1 169 33 _ 65 sinB = "n/I -cos 2 B = -\ /A „, 4 12 3 5 . cos(A + B) = 5 . 13 - 5 . 13 5 - 13 Example 6.19: Show that (i) sin(A + B) sin(A - B) = sin A - sin B (ii) cos(A + B) cos(A - B) = cos 2 A - sin 2 B sin(A +B) sin(A - B) = (sin A cosB + cosA sinB) (sinA cosB - cosA sinB) 2 2 2 2 = sin A cos B - cos A sin B = sin 2 A (1 - sin 2 B) - (1 - sin 2 A) sin 2 B = sin A - sin B cos(A + B) cos(A - B) = (cosA cosB - sinA sinB) (cosA cosB + sinA sinB) 2 2 2 2 = cos A cos B - sin A sin B = cos 2 A (1 - sin 2 B) - (1 - cos 2 A) sin 2 B 2 2 = cos A - sin B Example 6.20: If A + B = 45°, show that (1 + tanA) (1 + tanB) = 2 and hence deduce the value of tan 22 -^ Solution: Given A + B = 45° => tan (A + B) = tan45° 200 tanA + tanB 1 - tanA . tanB i.e. tanA + tan B = 1 - tanA . tanB i.e. 1 + tanA + tanB = 2 - tanA tanB (add 1 on both sides) 1 + tanA + tanB + tanA tanB = 2 i.e. (1 +tanA)(l + tanB) = 2 Take A = B then 2A = 45° => A = 22 ^ = B 1 + tan 22 2 J =2 => 1 + tan 22 ^ = ± V 2 .-. tan 22 2 = ±yJ2 - 1 Since 22 ~ is acute, tan 22 7 is positive and therefore tan 22 7 = "v2 - 1 Example 6.21: tan69° + tan66° . . tan (A - B) + tanB (i) Prove that - — : — ,„„ : — — = - 1 (n) - — : — — — 7— — 7 = tan A w 1 - tan69° tan66 1 - tan(A - B) tanB cos 17° + sinl7 Solution: tan69° + tan66° , rno rro ^ (1 ) 1 _ tan69 o tan66 o=tan(69° + 66°) (iii) T^Z • no = tan62° coslv -sinl7 = tan (135°) = tan (90° + 45°) = - cot45° = - 1 ,.. N tan (A - B) + tanB . „ N , (11) i — t 7T DW D = tan [(A - B) + B] = tanA 1 - tan(A - B) tanB LV ' J cos 17° + sinl7° (iii) L.H.S. = cosl7°-sinl7° Divide both Numerator and Denominator by cosl7° T TTO l+tanl7° tan 45° + tanl7° , _ „ LHS " = T^nT^ = l-tan45°tanl7° ( " tan45 = 1} = tan (45° + 17°) = tan62° = R.H.S. r n „ 1 In Example 6.22: Prove that (i) tan I 7 + 9 I tan 1 7 - 9 J = 1 1 7t (ii) If tanA = 3 and tanB = 7 , prove that A - B = 7 201 Solution: (i) L.H.S. = tan (7 + 9 J tan (7-8 1 + tanjn f 1 - tangj , ( 71 , r^nej Irr^neJ = ! l vtan 4 =1 3-1 $ tanA - tanB 2 2 7: (ii) tan(A-B)= 1+tanA tanB = — -^ =7 =1 = tan 1+3.2 2 71 71 tan (A - B) = tan 7 => A - B = 7 4 5 Example 6.23: If cos(a + (3) = 7 and sin (a - (3) = tt where (a + P) and (a - (3) are acute, find tan 2a. Solution: 4 3 cos (a + P) = 7 => tan (a + P) = 7 sin (a - P) = tt => tan (a - p) = 77 2a = (a + P) + (a - P) .-. tan2a = tan [(a + P) + (a - P)] 3 JL H tan (a + p) + tan(g - p) 4 + 12 _12 _ 56 ~~ 1 - tan(a + (3) . tan(a -p) _ , 3J_ _ jJ_~33 1_ 4 X 12 16 Example 6.24: Prove that tan3A - tan2A - tanA = tanA tan2A tan3A Solution: „ ^ tanA + tan2A tan3A = tan(A + 2A) = - — ; — — — 77 v ' 1 - tanA tan2A i.e. tan 3 A (1 - tanA tan2A) = tanA + tan2A i.e. tan3A - tanA tan2A tan3A = tanA + tan2A .-. tan3A - tan2A - tanA = tanA tan2A tan3A EXERCISE 6.4 (1) Find the values of (i) sin 15° (ii) cos 75° (iii) tan 75° (iv) sin 105° (2) Prove that (i) sin (45°+A) = —f= (sinA+cosA) (ii) cos(A+45°) = —fe (cosA-sinA) 202 (3) Prove that (i) sin (45° + A) - cos(45° + A) = ^2 sinA (ii) sin(30° + A) + sin(30° - A) = cosA (4) Prove that (i) cos(A + B) cos(A - B) = cos 2 B - sin 2 A (ii) sin(A + B) sin(A - B) = cos B - cos A (5) Prove that cos 2 15° + cos 2 45° + cos 2 75° = ^ (6) Prove that (i) sinA + sin(120° + A) + sin(240° + A) = (ii) cos A + cos(120° + A) + cos(120° - A) = (7) Show that (i) cosl5°-sinl5°=-7^ (ii) tanl5°+cotl5°=4 (iii) cot 75°+tan75° = 4 (8) (i) Find sin45° + sin30° and compare with sin 75° (ii) Find cos45° - cos30° and compare with cosl5°. (9) Show that (i) tan70° = 2 tan50° + tan 20° (ii) tan72° = tan 18° + 2tan54° (Hint : tanA tanB = 1 if A + B = 90°) cos 1 1 ° + sin 1 1 ° cos29° + sin29° (111) cosll°-sinll° = tan56 (1V) cos29°-sin29° = tan 74 nm p tv, t sin ( A ~ B ) A sin(B-C) A sin (C - A) (10) Prove that sinA sinB + sin B sinC + sin C sin A =0 5 1 (1 1) ( i) If tanA = g , tan B = yy show that A + B = 45° 1 1 % (ii) If tan a = y and tan P = t , show that a + (3 = t (12) If A + B = 45°, show that (cotA-1) (cotB - 1) = 2 and deduce the value of cot 22 ^ (13) If A + B + C = 7i, prove that (i) tanA + tanB + tanC = tanA tanB tanC (ii) tan2A + tan2B + tan2C = tan2A tan2B tan2C (14) If sinA = -~ , sinB = t find sin (A + B), where A and B are acute. (15) Prove that (i) sin (A + 60°) + sin(A - 60°) = sinA (ii) tan4A tan3A tanA + tan3A + tanA - tan4A = 203 6.3.2 Multiple angle identities: Identities involving sin2A, cos2A, tan3A etc. are called multiple angle identities. To develop these identities we shall use sum identities from the preceding lesson. We first develop an identity for sin2A. Consider sin (A + B) = sinA cosB + cosA sinB and put B = A sin2A = sin (A + A) = sinA cosA + cosA sinA = 2 sinA cosA Thus we have the identity | sin2A = 2sinA . cosA Identities involving cos2A and tan2A can be derived in much the same way as the identity above cos2A = cos (A + A) = cosA cosA - sinA sinA sin A cos2A = cos A Thus we have the identity Similarly we can derive The other useful identities for cos2A can easily be derived as follows: cos2A = cos 2 A - sin 2 A = (1 - sin 2 A) - sin 2 A y 2 2 cos2A = cos A - sin A 2tanA La tizA - t 1 - tan 2 A = 1 2sin 2 A cos2A = cos A - sin A = cos A - (1 - cos A) = 2cos 2 A - 1 From cos2A = 1 sin A = — 2sin A, also we have ■ cos2A Also, Hence cos2A = 2cos A - 1 2 1 + cos2A cos A = ~ cos2A 2, 1 tan A = 1 + cos2A sin2A = 2sinA cosA 2 sinA 7 2 tanA cos A = ■ cosA sec 2 A 2tanA 1 + tan 2 A 204 cos2A = cos A - sin A = cos A 1 y~ V cos A = cos 2 A(l -tan 2 A) 1 - tan 2 A 1 - tan 2 A sec 2 A 1 + tan 2 A Thus we have sin2A = 2 sinA . cosA 2 2 cos2A = cos A - sin A cos2A = 1 - 2sin A cos2A = 2 cos 2 A - tan2A = 2tanA 1 - tan 2 A sin2A = 2tanA 1 + tan 2 A cos2A = 1 - tan 2 A 1 + tan 2 A .3.3: Trigonometrical ratios of atios ofy sinA = sinl 2 x y = 2 sin y . cos y cos A = cos 12 xy 1 = = o 2 A i 2 cos y - 1 = 1 - 2 sin y tan A = tan ( 2 x y J 2 tany — 2 A 1 - tan y 2 A . 2 A s y - sin y 205 Similarly, we can prove the following identities A sinA = cosA = 2 tan "~ 1 + tan y 2 A 1 - tan y i 2A 1 + tan y . 2 A 1 - cosA in 2=~ 2 A 1 + cosA cos 2 - 2 2 A _ 1 - cosA tan 2 = 1 + cosA A sinA A 1 - cosA Also note that tany= 1+cosA and tan y= sinA 3 Example 6.25: If sin9 = 77 and 9 is acute, find sin29 ? Solution: sin9 = g ; cos 9 = \j 1 - sin 9 = A/l-^J = s~~ sin 29 = 2 sin9 cos9 = 2 . o . o - -,-, Example 6.26: Find (i) sinl5° (ii) tanl5° 64~8 3 ^/55 3a/55 o , . ,^ . ,„ • 39° / l-cos39 Solution : (l) sin 15 = sin^ - = v 9 (ii) tan 15° = tan -y sin30° 6.3.4 Trigonometrical ratios involving 3A sin3A = sin (2A + A) = sin2A . cosA + cos2A . sinA = 2 sinA cos 2 A + (1 - 2sin 2 A) sinA = 2sinA (1 - sin 2 A) + (1 - 2sin 2 A) sinA 206 = 3sinA - 4sin 3 A Similarly, cos3A = 4 cos A - 3cosA tan2A + tanA tan3A = tan (2A + A) = - — ; — rr— ; — - ■ ' 1 - tan2A . tanA 2 tanA "\ ^— + tanA 1 - tan 2 Ayl 2tanA 1 - tanA . ^~ 1 - tan 2 A _ 3 tanA - tan 3 A 1 - 3tan 2 A Example 6.27: Prove that cos A - sin A = cos2A Solution: L.H.S. = 1 . cos2A = cos2A = R.H.S. L.H.S. = (cos 2 A + sin 2 A) (cos 2 A - sin 2 A) Example 6.28: Solution: cot 3 A - 3cotA that cot3A = , 3cot 2 A - 1 1 3 „ cot 3 A-3cotA tan 3 A tanA R.H.S. — o — i 3cot 2 A - 1 3 j 1 - 3tan 2 A 3tanA - tan 3 A tan 2 A = —krr = cot3A = L.H.S. Example 6.29: 1 - cosB If tanA = — -j-g — , prove that tan2A = tanB, where A and B are acute angles. „ ■ 2B . B , _ 2sin Tr sinTr n 1 - cosB 2 2 B Solution : R.H.S = sinB = g g" = g = tan ^ 2sin 2 ■ cos 2 cos 7 B .". tan y = tanA 207 B ^A=2" =^> B = 2A Therefore tan2A = tanB Example 6.30: Show that 4 sinA sin (60° + A) . sin (60° - A) = sin3A Solution: L.H.S. = 4 sinA sin (60° + A) . sin (60° - A) = 4sinA { sin (60° + A) . sin (60° - A)} = 4sinA {sin 2 60 - sin 2 A} = 4sinA It - sin 2 Af = 3sinA - 4sin 3 A = sin3A = R.H.S. Example 6.31: Prove that cos20° cos40° cos80°=- L.H.S. = cos20° cos40° cos80° = cos20° {cos (60° - 20°) cos(60° + 20°)} = cos20° [cos 2 60° - sin 2 20°] = cos20° t - sin 2 20° = \ cos20° { 1 - 4(1 - cos 2 20°) } = T {4cos 3 20° - 3 cos20°} = 4 [cos3 x 20°] = 4 xcos60° = g = R.H.S. Example 6.32: Find the values of: (i)sinl8° (ii)cosl8° (iii) cos36° (iv) sin36° (v) sin54° (vi) cos54° Solution: (i) Let 9=18° then 59 = 90° => 29 = 90° - 39 => sin29 = sin(90° - 39) = cos39 =^> 2sin9 cos9 = 4cos 3 9 - 3cos9 => 2sin9 = 4cos 2 9 - 3 (.cos 9*0) => 2sin9 = 1 - 4sin 2 9 => 4sin 2 9 + 2sin9 -1=0 208 sin9 = -2+V4+16 - 1 ±-\/5 since sin 18° is positive, sinl8° = , (ii) cosl8° = V 1 - sin z 18 = A / 1 - P-^ (iii) cos36° = 1 - 2sin 2 18° = 4-^ (iv) sin36° = Vl-cos 2 36° = ^^ (v) sin54° = sin (90° - 36°) = cos36° = 4 — (vi) cos54° = cos (90° - 36°) = sin36° = ^ — j-^~ EXERCISE 6.5 (1) Prove the following: 1 n n 1 (i) 2sinl5° cosl5° = 2 (ii)sin-g cos 77 = 7—7^ (iii) sin72° = ^±M ( . v) cos72o = ^5_± , , 2 tan22 j° (v)l-2sin 2 22f=-j= (vi) — -p = 1 I - tan 22^ 71 (2) Show that 8 cos 75 - 6cos 77 = 1 n (3) If tan 2 = (2 - a/3) find the value of sin9 ^ N „ , 1 + sin 9 -cos 9 9 (4) Prove that : — 7 zr = tan x 1 + sin 9+ cos 9 2 (5) Prove that (i) cos 2 I 4 - 9 1 - sin 2 (7 - 9 1 = sin29 (ii) sec29+tan29 = tan [4 + 9 3 (6) (i) If tan9 = 3 find tan39 (ii) If sinA = 5 find sin3A 209 1 1 71 (7) If tana = t and tan (3 = y show that 2a + (3 = t (8) If 2 cos9 = x + — then prove that cos29 = 9 •* + ~ 6.3.5 Transformation of a product into a sum or difference We know that sin(A + B) = sinA cosB + cos A sinB ... (1) and sin(A - B) = sinA cosB - cosA sinB ... (2) Adding (1) and (2), we get sin(A + B) + sin (A - B) = 2 sinA cosB ... (I) Subtracting (2) from (1) sin(A + B) - sin(A - B) = 2 cosA sinB . . . (II) Again cos(A + B) = cosA cosB - sinA sinB ... (3) cos(A - B) = cosA cosB + sinA sinB ... (4) (3) + (4) => cos(A + B) + cos(A - B) = 2 cosA cosB . . . (Ill) (4) - (3) cos(A + B) - cos(A - B) = - 2sinA sinB . . . (IV) Now, let A + B = C and A - B = D then C+D C-D 2A = C + D (OR) A = —j— and 2B = C - D (OR) B = — ^— Putting these values of A and B in the above four formulae I, II, III and IV, we get ,, ■ ~ • - ~ • C + D C-D 1) sinC + sinD = 2 sin — ~ — • cos — 7 — „x ■ „ ■ ~ „ C + D . C-D 2) sinC - sinD = 2 cos — ~ — • sm — 7 — r. „ C + D C-D 3) cos C + cosD = 2 cos — ~ — • cos — 9 — C+D C-D 4) cosD - Cos C = 2sin — ~ — • s i n — 7 — Example 6.33: Express as sum or difference of following expressions, (i) 2sin29 . cos9 (ii) 2 cos29 cos9 (iii) 2 sin 3A . sinA 3A 5A (iv) cos79.cos59 (v)cos -j- -cos ~y~ (vi) cos39.sin29 (vii) 2cos3A . sin5A 210 Solution: (i) 2 sin 29 . cos9 = sin(29 + 9) + sin (29 - 9) = sin39 + sin9 (ii) 2 cos29 . cos9 = cos(29 + 9) + cos (29 - 9) = cos39 + cos9 (iii) 2 sin3A . sinA = cos(3A - A) - cos(3A + A) = cos2A - cos4A (iv) cos79 . cos59 = \ [cos(79 + 59) + cos(79 - 59)] = ^ [cos!29 + cos29] 3A 5A 1 (v) cos —j- . cos^ - = 2 3A 5A^ f3A 5A COS I ~y~ + ~7T I + COS i j ~ 1 = j [cos4A + cos(-A)] = 2 [cos4A + cosA] (vi) cos39 . sin29 =j [sin (39 + 29) -sin(39 - 29)] = j [sin 59 - sin 9] (vii) 2 cos3A . sin5A = sin(3A + 5A) - sin(3A - 5A) = sin8A - sin(-2A) = sin8A + sin2A Example 6.34: Express the following in the form of a product: (i) sin4A + sin2A (ii) sin5A - sin3A (iii) cos3A + cos7A (iv) cos2A - cos4A (v) cos60° - cos20° (vi) cos55° + sin55° Solution: • „ A • „ . „ ■ f4A + 2A\ f 4A - lK\ (I) sin4A + sin2A = 2 sin I ~ J cos I 9 1 = 2sin3A cosA ■ *. • o« „ r5A + 3A^ . f5A - 3A^ „ A A . A (II) sin5A - sin3A = 2cos I ~ J sin I 9 J = 2 cos4A sinA n. „ f3A + 7A\ (3 A - 7 A (III) cos3A + cos7A = 2 cos I ~ 1 cos I ir~ = 2cos5A cos (-2A)=2cos5A cos2A „ • f2A + 4M . f2A - 4A^ (iv) cos2A - cos4A = - 2sin I ~ 1 sin I ~ 1 = -2sin3A sin(-A)=2sin3A sinA (60° + 20°) f60° - 20°^ (v) cos60° - cos20° = - 2 sin i ^ sin I 2 J = " 2sin40 ° sin20 ° (vi) cos55° + sin55°=cos55° + cos(90°-55°) = cos55° + cos35° 55° + 35° 55° - 35° „ ien = 2 cos ^ cos ^ = 2cos45 coslO = 2-7= cosl0°=^/2 coslO 211 a/3 Example 6.35: Show that sin 20° sin40° sin80°=-|- Solution: L.H.S. = sin20 sin40° sin80°= sin20° ^ {cos40° -cosl20 c 1 __. f ._. 1 = 2 sin20° cos40° + 2 = 2 sin20° cos40° + 4 sin20° = 4 (sin60° - sin20°) + 4 sin20° = 4 sin60° = f =R.H.S. Example 6.36: Prove that 4(cos6° + sin24°) = ^3 + -y/B Solution: 4(cos6° + sin24°) = 4 (sin84° + sin24°) [v cos6° = cos(90°-84) = sin84°] „ „ . (84° + 24°^ f84° - 24°' = 4 . 2sin ~ cos 2 ; wo v 2 V5 + 1 = 8 sin54° . cos30° = 8 = >/l5 +^3 Example 6.37: Prove that (i) cos20°+cosl00° + cosl40° = (ii) sin50 o -sin70°+sinl0°= Solution: (i) L.H.S. = cos20° + (cosl00° + cosl40°) ™ n „ ( lOQ + 140°^ flOO-140' " = cos20 + 2cos I 2 1 • cos I 7 = cos20° + 2cosl20° cos(- 20°) = cos20° + 2[- 2) cos20° = cos20° - cos20° = = R.H.S. (ii) L.H.S. = sin50° - sin70° + sinl0° f50 + 70°^ . f50 - 70°^ . ,„ n = 2 cos ~ .sin ~ + sinlO 212 = 2 cos60° . sin(- 10°) + sinl0° = 2x^ (- sinl0°) + sinl0° = - sinl0° + sinl0° = = R.H.S. 6.3.6 Conditional Identities Example 6.38: If A + B + C = 7i, prove that sin2A + sin2B + sin2C = 4sinA sinB sinC Solution: L.H.S. = sin2A + sin2B + sin2C = (sin2A + sin2B) + sin2C = 2sin(A + B) cos(A - B) + sin2C = 2sin(Tt - C) cos(A - B) + sin2C = 2sinC cos(A - B) + 2sinC cosC = 2sinC {cos(A - B) + cosC} I = 2 sinC lcos(A - B) + cos(180 - A + B)J = 2 sinC {cos(A - B) - cos(A + B)} = 2sinC {2 sinA sinB} = 4 sinA sinB sinC = R.H.S. Example 6.39: If A + B + C = 180° Prove that cos2A + cos2B-cos2C = l-4sinA sinB cosC Solution: L.H.S. = cos2A + (cos2B - cos2C) = 1 - 2sin 2 A + { "2 sin(B + C) sin(B - C) } = 1 - 2sin 2 A - 2sin(180° - A) sin(B - C) = 1 - 2sin 2 A - 2sinA sin(B - C) = 1-2 sinA [sinA + sin (B - Q] = 1 - 2sinA [sin(B + C) + sin(B - C)] , [v A = 180° - (B + C)] = 1 - 2sinA [2sinB cosC] = 1 - 4sinA sinB cosC = R.H.S. Example 6.40: If A+B+C = %, prove that cos 2 A + cos 2 B - cos 2 C = 1 - 2sinA sinB cosC Solution: L.H.S. = cos 2 A + cos 2 B - cos 2 C = (1 - sin 2 A) + cos 2 B - cos 2 C = 1 + (cos 2 B - sin 2 A) - cos 2 C = 1 + cos(A + B) . cos(A - B) - cos 2 C = 1 + cos(tt - C) cos(A - B) - cos C = 1 - cosC. cos(A - B) - cos 2 C 213 = 1 - cosC [cos(A - B) + cosC] = 1 - cosC [cos(A - B) -cos (A + B)] = 1 - cosC [2sin A sinB] = 1 - 2sinA sinB cosC = R.H.S. EXERCISE 6.6 (1) Express in the form of a sum or difference: (i) 2sin49 cos29 (ii) 2cos89 cos69 (iii) 2cos79 sin39 (iv) 2sin3A sinA (v) 2cos6A sin3A (vi) cos49 sin99 3A A 7A 5A 59 49 (vii) cos ~y- sin y (viii) sin~y~ cos ~y- (ix) cos y cos y (2) Express in the form of a product: (i) sinl3A + sin5A (ii) sinl3A - sin5A (iii) cosl3A + cos5A (iv)cosl3A-cos5A (v) sin52° - sin32° (vi) cos 51° + cos23° (vii) sin80° - cos70° (viii) sin50° + cos80° (ix) sin20° + cos50° (x) cos35° + sin72° 3 (3) Prove that sin20° sin40° sin60° sin80°=Yg (4) Prove that cos20° cos40° cos60° cos80 o = yg (5) Prove that sin50° - sin70° + cos80° = ? 9 9 [ Ct + 6 (6) Prove that (cosa + cos(3) + (sina - sin(3) =4cos I — y^ ,„ _ , _ sin3A-sinA _ A ,.. cos2A -cos3A A (7) Prove that (l) : r— = cot2A (n) • - . , — • - . = tan tt ' ' cosA - cos3A v ' sin2A + sin3A 2 (8) A + B + C = 7t, prove that sin2A - sin2B + sin2C = 4 cosA sinB cosC (9) If A + B + C=180°, 2 2 2 prove that sin A + sin B + sin C = 2 + 2cosA cosB cosC A B B C C A (10) If A+B+C = 7i, prove that tan y tan y +tan y tan y +tan y tan y = 1 „.„, T „ . „ „„„ , , sin2A + sin2B + sin2C (11) If A + B + C = 90°, show that — —. r^r~ —- = cot A cot B sin2A + sin2B - sin2C 2 A 2 B 2C (12) Prove that A + B + C = 71, prove that sin y + sin y + sin y 1 o • A • B • C = 1-2 siny sin y sin y 214 6.4 Trigonometrical Equations An equation involving trigonometrical function is called a trigonometrical equation. 1 2 1 cos9 = ;y> tan 9 = 0' cos 9 ~~ 2sin9 = y are some examples for trigonometrical equations. To solve these equations we find all replacements for the variable 9 that make the equations true. A solution of a trigonometrical equation is the value of the unknown angle that satisfies the equation. A trigonometrical equation may have infinite number of solutions. The solution in which the absolute value of the angle is the least is called principal solution. Note that trigonometrical equations are different from trigonometrical identities. It is possible that some equations may not have solution. For example cos9 = 4 has no solution. The expression involving integer V which gives all solutions of a trigonometrical equation is called the general solution. 6.4.1 General solutions of sin 9 = ; cos9 = ; tan 9 = Consider the unit circle with centre at O(0, 0) Let a revolving line OP, starting from OX, trace |XOP =9 Draw PM perpendicular to OX (l)sin9 = tY In the right angled triangle OMP we have OP = 1 unit, MP sin9 = OP sin9 = MP If sin9 = 0, then MP = 0, i.e. OP coincides with OX or OX' V Fig. 6.20 .'. |XOP = 9 = 0, 7t, 2tc, 3tc, ... [in the anti clockwise direction] or 9 = - 7i, - 2tc, - 3tc, [in the clockwise direction] i.e. 9 = or any + ve or - ve integral multiple of n. Hence the general solution of sin9 = is given by 9 = nn, n e Z, where Z is the set of all integers. (2) cos9 =0 In the right angled triangle OMP we have cos9 = ~Qt7 = OM (.OP = 1 unit) 215 If cosG = 0, then OM = i.e. OP coincides with OY or OY' i.e. 9 = 2 > "t~ > 5y , [in anticlockwise direction) . tc 3tc 5n . or 9 = - 2 » _ "t" . _ "t" [m clockwise direction] solution which is in 71 71 1 for sine, in I - y, 9" J f° r tangent and in [0, Tt] for i.e. 9 = + I odd multiple of ~ Hence the general value of 9 is given by 9 = (2« +1)9" ,neZ (3) tan9 = MP In the right angled triangle OMP, if tan9 = then q^ = or MP = Proceeding as in (1), we get 9 = tin, n e Z Thus, (1) If sin9 = 0, 9 = tin, neZ (2) If cos9 = 0, 9 = (2«+l)| , weZ (3)Iftan9 = 0, 9 = mtc, weZ When a trigonometrical equation is solved, among all solutions the -71 71 2 ' 2 cosine, are the principal values of those functions. Example 6.41: Find the principal value of the following : (i) cosx = 2 (ii) cos 9 = - 2 (***) cosec9 = - "7^ (iv) cot9 = - 1 (v) tan9 = ^3 Solution: (i) cos* = 2 > .". x lies in the first or fourth quadrant. Principal value of x must be in [0, Tt]. Since cos* is positive the principal value is in the first quadrant cosx = 2 = cos a an ° a e [0> TC ] n :. The principal value of x is t . 216 a/3 (ii) cos9 = -^- <0 Since cos 9 is negative, 9 lies in the second or third quadrant. But the principal value must be in [0, n] i.e. in first or second quadrant. The principal value is in the second quadrant. .-. cos9 = - ^r = cos (180° - 30°) = cosl50°. 5n The principal value is 9 = 150° = ~~z~ . (iii) cosec9 = - ~~7^ => sin9 = - 2 < .". 9 lies in the third or fourth quadrant. But principal value must be in n % 2' 2_ n i.e. in first or fourth quadrant. .\ 9 = - t (iv) cot9 = - 1 .-. tan9 = - 1 < ( n n :. 9 is in the second or fourth quadrant. Principal value of 9 is in I - t", 9 .". the solution is in the fourth quadrant. f n] . . n ( n n cot {-4) =~ l => e = "4 e l"2' 2 6.4.2 General solutions of sin 9 = sin a ; cos 9 = cos a ; tan 9 = tan a (1) sin9 = sina -y ^a< y ie - a ^> sin9 - sina = '2' 2 9 + a] . fQ-a 2 cos 1 — j — j . sin 1 — ^ — I = 9 + a\ „ .fQ-a cos I — ^ — I =0 or sin I — ~" — I = 9 + a 7t 9 - a — ~ — = (2m + 1) y , or — j — = tin, n e Z 9 + a = odd multiple of n or 9 - a = even multiple of n 9 = (odd multiple of n) - a . . . (1) or 9 = (even multiple of n) + a . . .(2) 217 Combining (1) and (2), we have 9 = tin + (- l)".a, where n e Z (2) cos9 = cosa < a < n i.e. a e [0, n] => cos9 - cosa = 9 + a\ . (Q-a ■ 2sin I — j — I • sin I — n — I = 9 + a\ „ . (Q - a sin I — 2 — I = or sin[ — ^ — | = 9 + a „ 9 - a „ => — j — = tin ; n€ Z or — ~ — = tin ; n e Z => 9 = 2wtc - a or 9 = 2«7T + a Hence Q = 2nn±a ' n eZ. (3) tan9 = tana - y < a < t i.e. a e I - x, 7 sin9 sina cos9 _ cosa => sin9 cosa - cos9 sina = => sin (9 - a) = => 9 - a = tin, n e Z => 9 = wrc + a, « eZ Thus, we have sin 9 = sina =>9 = ra + (-lfa;/i6Z cos 9 = cos a => 9 = 2wtc + a ; n e Z tan 9 = tan a => 9 = wTC + a;weZ Example 6.42: Find the general solution of the following : (i) sin9 = 2 (ii) sec9 = - ^2 (iii) cos 2 9 = 4 (iv) cot 2 9 = 3 (v) sec 2 9 = j Solution: (i) sin9 = « 1 tc n sin 9 = 9 = sin t which is of the form sin9 = sina where a = ~z n .'. The general solution is 9 = tin + (- 1)" . ~z ', n e Z (ii) sec9 = -^J2 =^> cos 9 = - -75- < Principal value of 9 lies in [0, n] As cos 9 is negative, the principal value of 9 lies in second quadrant. 218 3n f it] n I COS "J" = COS I 7T - 4 I = - COS 4 = - ~jE :. 9 = 2nit ± ~r ; n e Z (iii) We know that cos29 = 2cos 9-1 A . 1 , 1 it 2it = 2 It I -1=t-1=-t=- cos t = cos : 2u .". 29 = 2mt + ~t~ ;neZ 9 = «jr + T ; n e Z 22 2 (iv) We know that 1 + cot 9 = cosec 9 => 1 + 3 = cosec 9 2 2 1 .". cosec 9 = 4 or sin 9 = T cos29 = 1-2 sin 9=1- 2l tJ = t = cos t 71 .". 29 = 2mt ± t ; « e Z 9 = Mjr + T ;weZ 2 2 4 1 (v) We know that tan z 9 = sec z 9- 1 = t -1=3 l-tan 2 9 cos 29 = t = 1 + tan z 9 .-I .4 2 3 _ 4 3 1 ~2 COS 29 = T = COS T 29 = 2nit ± t ; neZ 9 = M7r + T ; n e Z Note : Solve : sin9 = ' ^ 7T 4rt There are two solutions in < 9 < 2% i.e. 9 = - t and ~t~ 219 The general solution is 71 9 = wi + (-l) n l-Tj ; n eZ ...(1) Even if we take 9 = n n + (- 1)" I ~r~ J ;neZ . . . (2) The solution will be the same although these two structures are different. Here the solution sets of (1) and (2) are same. But the order in which they occur are different. 4ti For example Put n = 1 in (1), we get, = ~t" 4n Put n = in (2), we get, 9 = ~r~ It is a convention to take that value of 9 whose absolute value is least as a (principal value) to define the general solution. Example 6.43: Solve : 2cos 9 + 3sin9 = Solution: 2cos 2 9 + 3sin9 = => 2 (1 - sin 2 9) + 3 sin9 = => 2sin 2 9 - 3sin9 - 2 = => (2 sin9 + 1) (sin 9 - 2) = => sin 9 = ~y~ (v sin9 = 2 is not possible) => sin 9 = - sin r =^> sin 9 = sin I - t ^ e = n7t + (-lf.f-|) ; weZ Example 6.44: Solve : 2tan9 - cot9 = - 1 Solution: 2 tan9 - cot9 = - 1 2 tan 9 - — "r = - 1 tan9 ^•2tan 2 9 + tan9-l =0 (2 tan9 - l)(tan9 + 1) = 220 2 tanG -1=0 or tanO +1=0 tan9 = 2 or tan 9 = - 1 n When tan9 = - 1 = - tan t tanG = tan | - t =>Q = nn + \-~7 = nn-T ; n e Z When tan 9 = ~ = tan(3 (say) .-. 9 = nn + P -lf r = nn + tan I ~ Hence 9 = nn - t or 9 = rat + tan" [ ~ J ', n e Z Example 6.45: Solve : sin2x + sin6x + sin4x = Solution: sin2x + sin6x + sin4x = or (sin6x + sin2x) + sin4x = or 2sin4x . cos2x + sin4x = sin4x (2 cos2x + 1) = nn when sin4x = => 4x=nn or x = ~r ; we Z _ _ — 1 n ( n] In When 2 cos2x + 1 = => cos 2x = ~y = - cos t = cos I n — w I = cos ~r~ 27r n .". 2x = 2nn ± ~T" or x = n n ± t Hence x = ~r~ or x = nn±^ ; n e Z Example 6.46: Solve : 2sin jc + sin 2x = 2 2 2 Solution: 2 sin x + sin 2x = 2 2 2 .". sin 2x = 2- 2sin * 2 = 2(1 - sin x) 2 2 sin 2x = 2 cos x 221 2 2 2 4sin x cos x - 2 cos x = 2 2 2 2(1 - cos x) cos x - cos x = 2cos x - cos x = 2 2 cos x (2 cos x - 1) = cos x = 2 2^ cos x = cos 2 => x = nn +^ , n & Z COS X = 2 = '-%f 2 2 n COS X = COS T x = mn ± T , m e Z Example 6.47: Solve : tan 2 9 + (l - a/3) tan9 - a/3 = =^> tan 2 9 + tan9 - a/3 tanG -a/3 =0 => tanG (tan9 + 1) - a/3 (tanG + 1) = =} (tanG +1) (tanG -a/3 ) = =^> tanG = - 1 tanG = tan| - j G = nn ■ neZ tanG = a/3 tanG = tan t mn + T , m e Z 6.4.3 Solving equation of the form a cosG + £> sinG = c. where c <a +b acosG + bsinG = c ■••(!) Divide each term by ^a + b , la 2 + b 2 cosG + V Choose cosa = sinG = V« 2 + b 2 V a 2 + b 2 sina = ' ^\ja +b \ja +b :. (1) becomes cosG cosa + sinG sina = cos(3 => cos (G - a) = cos(3 =^> G - a = 2nn ± (3 => G = 2nn + a + (3 , n e Z Example 6.48: Solve : a/3 sin x + cosx = 2 and cos(3 = V 2 , i2 222 2 2 2 Solution: This is of the form a cos* + b sinx = c, where c < a + b So dividing the equation by a/(a/3~) +1 or 2 „ T a/3 . 1 , . 71 TC We get ^ sin x + j coax =1 => sin t . sinx + cos t . cos x = 1 i.e. cosIjc — Tl = 1 cos i x - t I = cos x — T = 2mtc + i.e. x = 2wrc + T , « e Z EXERCISE 6.7 (1) Find the principal value of the following equations: 1 (i) sin9 = V2 (ii) 2 cos9 -1=0 (iii) a/3 cot 6 = 1 1 i/1 ' 2 (iv) a/3 sec9 = 2 (v) sinx = (vii) sec x = 2 (2) Find the general solution of the following equation: (vi) tan9 = 1 (i) sin29 = ^ (ii) tan9 = - a/3 (iii) cos 39 = V3 V2 (3) Solve the following : (i) sin3x = sinx (ii) sin Ax + sin2x = (iii) tan2x = tanx (4) Solve the following: ? 1 (i) sin 2 9 - 2cos9 + 4=0 (iii) cosx + cos2x + cos3x = (5) Solve the following: (i) sin9 + cos9 = a/2 (iii) ^2 sec9 + tan9 = 1 2 2 (ii) cos x + sin x + cosx = (iv) sin2x + sin4x = 2sin3x (ii) sin9 - cos9 = - a/2 (iv) cosec9 - cot9 = a/3 223 6.5 Properties of Triangles Consider a triangle ABC. It has three angles A, B and C. The sides opposite to the angles A, B, C are denoted by the corresponding small letters a, b, c respectively. Thus a = BC, b = CA, c = AB. Fig. 6.21 We can establish number of formulae connecting these three angles and sides. I. Sine formula: In any triangle ABC, ^^ sin B ~~ sinC of the circum circle of the triangle ABC. In fig(6.22) O is the circumcentre of the triangle ABC. R is the radius of the circumcircle. Draw OD a perpendicular to BC. Now BC = a, BD = ~ Clearly ABOC is an isosceles triangle. We know that |BOC = 2 |BAC = 2A .-. I BOD =A = 2R. Where R is the radius Fig. 6.22 From the right angled triangle BOD, Similarly, we can prove II. Napier's formulae In any triangle ABC (1) tan (2) tan sinA = BD all = R _ R _ a 2R 2R sinA = a -- a or sinA = 2R b sinB = sinC = 2R a b c sinA (3) tan" A-B 2 B-C 2 C-A sinB a — b a + b b - c b + c sinC = 2R C cot cot 2 c a B — ; — cot 7T are true c + a 1 These are called Napier's formulae 224 Result (1): tan - A-B a-b a + 1 C coty Proof: From sine formulae a-b a + b C coty = 2R sinA - 2R sinB C 2R sinA + 2RsinB cot 2 .•. tan = sinA - sinB C sinA + sinB cot 2 = A+B . A-B 2 COS r. Sill r. _ „ . A + B A-B uuL 2 2 sin j cos ~ = fA + B\ A-B C cot 1 2 J tan 2 cot 2 = f™ °l A " B c cot 1 90 - 2 1 tan 2 cot 9 = C A-B C A-B tan 2 tan ^ cot 2 = tan ^ A- -B a-b C cot 2 - a + b " Ul 2 Similarly, we can prove other two results (2) and (3) III. Cosine formulae In any triangle ABC, the following results are true with usual notation Results: (1) a 2 = b 2 + c 2 - 2bc cosA (2) b 2 = c 2 + a 2 - lea cosB (3) c 2 = a 2 + b 2 - lab cosC These are called cosine formulae Result (1): a 2 = b 2 + c 2 - 2bc cos A Proof: Draw CD perpendicular to AB. Now a 2 = BC 2 = CD 2 + BD 2 = (AC 2 - AD 2 ) + (AB - AD) 2 = AC 2 - AD 2 + AB 2 + AD 2 - 2AB x AD = AC 2 + AB 2 - 2AB x (AC cosA) 2 2 2 a = b + c - 2bc cosA Fig. 6.23 225 Similarly we can prove the other results (2) and (3) We can rewrite the formulae in different formats. 2 2 2 ,2 ,2 2 ? + c cosA = cosB = ' 2 2 c + a ■ lea cosC = 2,2 2 a + b - c lab IFF IV. Projection formulae In any triangle ABC (1) a = b cos C + c cosB (2)b = c cosA + a cosC (3) c = a cosB + b cosA are true with usual notations and these are called projection formulae. Result (1): a = b cosC + c cosB Proof: In triangle ABC, draw AD perpendicular to BC. From the right angled triangles ABD and ADC, BD cosB = cosC = But AB DC AC BD = AB x cosB DC = AC x cosC Fig. 6.24 BC = BD + DC = AB cosB + AC cosC a = c cosB + b cosC or a = b cosC + c cosB Similarly, we can prove the other formulae (2) and (3) V. Sub-multiple (half) angle formulae In any triangle ABC, the following results are true. „. . A . j(s-b)(s-c) (1) sin 2 =-\j bc (2)sin| =-\ (4) cos y = ^ (6) cos y = ' (8) tan 2" =" /(j -c) (s - a) J ca ,,, . C . I(s-a)(s-b) (3) sin 2=^/ ab s(s - a) B Is(s-b) (5)cos 2 =^/ \ a /j(j - c) ,-,, A . j(s-b)(s-c) (7) tan 9 = \ / x 2 \j s(s- a) , m , C . j(s-a)(s-b) (9) tan 9 = A / 2 \j s(s- c) a + b + c where j - ,-, /(i 1 — c) (5 — a) V s(j - b) The above results are called sub-multiple angles (or half angle) formulae. 226 Result (1): siny = V 1 ^ c) Proof : We know that cos2A = 1 - - 2sin 2 A 2sin 2 A = 1 - - cos2A Replacing A by y , 2 sin "2=1- -cosA = 1- u2 , 2 b + c - 2bc 2 a a 2 -(b-c) 2 2bc - b - c + a 2bc (a + b - c) (a - b + c) 2bc ~ 2bc (a + b + c -2c)(a + b + c - 2b) 2bc (2s - 2c) (2s - 2b) 2bc . . 2 A 2(s-c)2(s-b) 2sin 2 = 2^ 2 A (s-b)(s-c) sin 2 = fc a + b + c = 2s sin y = ±M ^ Since y is acute, sin y is always positive. , . . (s-b)(s-c) rhus sin 2=V te Similarly we can prove the other two sine related formulae (2) and (3) Result (4): cos y = M ^ Proof : We know that cos2A = 2cos A - 1 2cos 2 A = 1 + cos2A A T A Replacing A by y , 2 cos y = 1 + cosA b + c - a 2bc + b + c - 2bc ~ 2bc (b + c) 2 -a 2 (b + c + a)(b + c - 2 - a - a) 2bc ~ 2bc 227 2 cos cos (b + c + a) (b + c + a - 2a) 2s(2s - 2a) 2bc ~ 2bc 2s x 2{s — a) Result (7): Proof: cosy = tany = tany = 2bc s(s — a) be js(s-a) o cosine related formulae ( . ks-b)(s- c) \j s(s - a) A l(s - i) (s - c) sin 2 -\J £>c cosy ^ / (j-fr)(j-c) j(j - a) Similarly we can prove other two tangent related formulae (8) and (9) VI. Area formulae (A denotes area of a triangle) In any triangle ABC (l)A = 2<a*sinC (2) A = ~ be sinA (3) A = ~ ca sinB (4)A = ^ (5) A = 2R 2 sinA sinB sinC (6) A = a/^ - a) (s - b) (s - c) are true with the usual notations and these are called Area formulae. Result (1): A = ~ ab sinC Proof : Draw AD perpendicular to BC A = Area of triangle ABC 1 1 = 2 x BC x AD = 2 x BC x AC x sinC 1 AD = 2®b sinC [v sinC = ~ttt => AD = AC x sinC] Similarly we can prove the results (2) and (3) Fig. 6.25 228 abc Result (4): A = ^- Proof: We know that A = 2 ab sinC 1 u c - 2 cid 2R abc = 4R C =2R sinC Result (5): A = 2R Z sin A sinB sinC Proof: We know that A = ~~ ab sinC = ~ 2R sinA 2R sinB sinC ~:a = 2R sinA = 2R 2 sinA sinB sinC b = 2R sinB Result (6) Prove that A = ^js(s - a) (s - b) (s - c) Proof: We know that A = x ab sinC 1 C C = y ab 2sin y cos y = ^i'Ci' - a) (s - b) (s - c) Example 6.49: In a triangle ABC prove that a sinA - b sinB = c sin(A - B) Solution: By sine formulae we have a b c sinA = sinB = sinC = .". a = 2R sinA, b = 2R sinB, c = 2R sin C a sinA - & sinB = 2RsinA sinA - 2R sinB sinB = 2R (sin 2 A - sin 2 B) = 2R sin(A + B) sin(A - B) = 2Rsin(180-C)sin(A-B) 229 = 2R sinC sin(A - B) = c sin (A - B) , ,«, , sin(A - B) a 2 -b 2 Example 6.50: Prove that • , » r,*. = 5 — Solution: By sine formula ^ = = = ^ = 2R a 2 -fr 2 (2R sinA) 2 - (2R sinB) 2 c 1 ~ (2R sinC) 2 4R 2 sin 2 A - 4R 2 sin 2 B sin 2 A - sin 2 B 4R 2 sin 2 C sin 2 C sin(A + B) sin(A - -B) [sinC = sii sin 2 C sin(A + B) sin(A - -B) sin(A - B) sin 2 (A +B) ~ sin(A + B) Example 6.51: Prove that X a sin (B - C) = Solution: S a sin (B-C) = a sin(B -C) + b sin(C - A) + c sin (A - B) = 2R sinA sin(B-C) + 2R sinB sin(C - A) + 2R sinC sin(A - B) sinA = sin(B + C), sinB = sin(C + A) ; sinC = sin(A + B) = 2R sin(B + C) sin(B - C) + 2R sin(C + A) sin(C - A) + 2R sin(A +B) sin(A - B) = 2R [sin 2 B - sin 2 C + sin 2 C - sin 2 A + sin 2 A - sin 2 B] = example 0.: •>£: riove mat cos j = sin j Solution: b + c . A 2R sinB + 2R sinC . A sin r, = <->r> • a sin a 2 2R sinA 2 sinB + sinC A = sinA sin 2 „ . B + C B-C 2 sin 2 cus 1 a A A sm 2 2 sin y cos y 230 . B+C B-C sin ~ cos ^ A cosy . fl80 -fC\ B sin 1 2 1 cus -C 2 A cosy • (— A^ B- C sin 1 90 — j 1 cos j A cosy B-C = cos 2 .• sin (90 -f) A = cosy Example 6.53: In any triangle ABC prove that a 2 sin (B - sinA - C) b 2 sin(C - A) c x sinB x 2 sin(A - sinC B) Solution: 2 a sin (B -Q (2R sinA) 2 sin(B - C) 4R 2 sin ; 2 A sin (B- -C) sinA — sinA sinA = 4R 2 sinA sin(B - C) = 4R 2 sin(B + C) sin(B - C) = 4R 2 (sin 2 B - sin 2 C) = 4R 2 sin 2 B - 4R 2 sin 2 C = b 2 -c 2 2 sin(C - A) 2 2 Similarly ^g = c - a c 2 sin(A - B) 2 2 sh^ =fl "^ a 2 sin (B - C) b 2 sin(C - A) c 2 sin(A - B) sinA ' sinB ' sinC 1? 2,2 2,2,2 = b-c+c-a+a-b = EXERCISE 6.8 In any triangle ABC prove that 9 9 9 A 9 9 A (1) a = (b + c) sin y + (b - c) cos y 231 (2) X a(b 2 + c 2 ) cosA = 3abc (3) S a(sinB - sinC) = (4) X (b + c) cosA = a + b + c (5) a 3 sin(B - C) + b 3 sin(C - A) + c 3 sin(A - B) = (6) a(fc cosC - c cosB) = b 2 - c 2 2 2 2 cosA cos B cos C a + b + c 2abc a tanA + b + 2 , 2 c + a - -£ 2 tanB ~ j2 , 2 + c - 2 - a (7) (8) (9) If a cosA = b cosB then show that the triangle is either an isosceles triangle or right angled triangle? 6.6. Solution of triangles We know that a triangle has six parts (or elements). Consider a triangle ABC. With usual symbols, the sides a, b, c and the angles A, B, C are parts of the triangle ABC. The process of finding the unknown parts of a triangle is called the solution of triangle. If three parts of a triangle (atleast one of which is a side) are given then the other parts can be found. Here, we shall discuss the following three types. 1) Any three sides (SSS) are given. Fig. 6.26 2) Any one side and two angles (SAA) are given. 3) Any two sides and the included angle (SAS) are given. Type I: Given three sides (SSS) To solve this type, we can use any one of the following formulae. (a) Cosine formula (b) Sine formula (c) Half angle formula . It is better to use cosine formula if the sides are small, while use half angle formula if the sides are large. Example 6.54: Given a = 8, b = 9 s c = 10, find all the angles. Solution: To find A, use the formula 2 a = 2 2 b + c - 2bc cosA cos A = b 2 + c 2 -a 2 81 + 100- 2bc ~ 180 -64 117 _ 180 A = 49° 28' 232 Similarly cosB = c 2 + a 2 -b 2 100 + 64-81 160 83 160 2ca B= 58° 51' A + B + C= 180° .-. C = 180° -(49° 28' + 58° 51') = 71° 41' A = 49° 28', B = 58° 51', C = 71° 41' Note: In the above example the numbers are smaller and hence we used cosine formula. Example 6.55: Given a = 31, b = 42, c = 57, find all the angles. Solution: Since the sides are larger quantities, use half angle formulae a + b + c But Thus s = = 65 log tan" A A' tan y = 1 (s-b)(s-c) = ^23x8^ 2 s(s - a) 1,65 x 34j 2 [log23 + log8 - log65 - log34] 2 [1-3617 + 0. 9031 - 1.8129 - 1.5315] \ [-1.0796] = \ [-2 + 0.9204] = \ I 2 + 0.9204J = 1 , 4602 Y = 16°6'^A = 32° 12' B tan j = (s - c) (s - a) s(s - b) 1 65 x 23 J lOE B' tan 2 = 2 [ lo g 8 + lo S 34 " lo S 65 " lo S 23 ] = 2" [-0.7400] = 2" [-2+1.2600] 2 + 1.2600. = 1 .6300 233 => 2 = 23 ° 6 ' => B = 46 ° 12 ' C= 180-(A + B) = 101°36' Thus A= 32° 12' B = 46° 12' C=101°36' Type II: Given one side and any two angles (SAA) To solve this type, draw a sketch of the triangle roughly, for better understanding and use sine formula. Example 6.56: In a triangle ABC, A = 35° 17', C = 45° 13', b = 42.1. Solve the triangle. Solution: The unknown parts are B, a, c B = 180 - (A + C) = 180 - (35° 17' + 45° 13') = 99° 30' To find the sides, use sine formula sinA sinB sinC 6sinA 42.1 xsin35° 17' a = Again sinB s in99° 30' log a = log 42.1 + log sin35° 17' - log sin99° 30' = 1.6243+ 1 .7616 - 1 .9940 = 1.3859- 1.9940 = 1.3859 -[-1 + 0.9940] = 1.3919 a = 24.65 frsinC _ 42.1 xsin45° 13' c ~ sinB ~ sin99'30' logc = log 42.1 +logsin45° 13' -log sin99° 30' = 1.6243+ 1 .8511 - 1 .9940 Thus = 1.4754- 1 .9940 = 1.4754 -[-1+0.9940] = 1.4814 c = 30.3 B = 99° 30', a = 24.65, c = 30.3 234 Type III: Given two sides and the included angle (SAS) Since two sides and the included angle are given, the third side can be found by using the proper cosine formula. Then one can apply the sine formula to calculate the other elements. Example 6.57: Solve the triangle ABC if a = 5, b = 4 and C = 68°. 2 i 2 2 2 2 Solution: To find c, use c = a + b - lab cosC = 25 + 16 - 2 x 5 x 4 cos 68° = 41-40x0.3746 = 26.016 c = 5.1 To find the other two angles, use sine formula. b sinC 4 x sin68° => sinB = ^^=—^ l log sinB = log 4 + log sin68° -log 5.1 = 0.6021 + 1 .9672 - .7075 = 0.5693-0.7075 = -0.1382 = 1 .8618 B = 46° 40' => A = 180 -(B + C)=180 -(114° 40') = 65° 20' Thus B = 46° 40', A = 65° 20', c = 5.1 Note: To find the angles A and B one can also use the tangent formula A-B a-b C tan — ~ — = — ~r cot^r 2 a + b 2 6.7 Inverse Trigonometrical functions (Inverse circular functions) The quantities sin - x, cos - x, tan - x, ... are called inverse circular functions, sin - x is an angle 9, whose sine is x. Similarly cos - x denotes an angle whose cosine is x and so on. The principal value of an inverse function is that value of the general value which is numerically the least. It may be positive or negative. When there are two values, one is positive and the other is negative such that they are numerically equal, then the principal value is the positive one. _i fl\ n n For example the principal values of cos I y I is t and not - t though 3 n\ 1 COS |— a I - 9 235 Note : sin - x is different from (sinx) ~ . sin - in sin - x denotes the inverse of the circular function. But (sinx) - is the reciprocal of sinx i.e. -^— . The Domain and Range of Inverse Trigonometrical functions are given below: Function Domain Range (Principal Value) 1. y = sin - x -1<JC<1 n n 2. y = cos x -1<X<1 0<y<n 3. y = tan - x R % n -2<y<2 4. y = cosec - x x > 1 or i < - 1 - 2 ^ y ^ 2 ' y * ° 5. y = sec - x x > 1 or x < - 1 n 0<y<n; y^2 6. y = cot x R 0<y<n Table 6.6 Example 6.58: Find the principal values of: (i)Sin -1 (£) (ii)sec -1 ^ (hi) tan -1 (-4 (v) cos - - 9 (vi) cosec - (- 2) (iv) sin -1 (- 1) Solution: (i) Then • -l f^\ ~ n n Let sin lyl = y* where ~j- <y< -~ sm Ml 1 = y => sin y = 2 = sin" y=e :. The principal value of sin I ~ J is ~z (ii) Let sec —j= = y, where < y < w , then, sec ~ l H) = y^»~y = j- 3 :. The principal value of sec 2 n n = y => sec y = 77^ = sec g => v = g _1 U IS' 236 -1 ( 1 ^ , n n Let tan - —j= = y, where - — < y < — (iii) lcl ian i - r- Then tan -1 (- 77=) = y => tarry = - -^ = tan (- |J ^y = -^ :. The principal values of tan" - ~7f is -t~~ (iv) Let sin (-1) = y, where —~~ < x < ~~ Then, sin" (-1) = y => siny = - 1 -1 = sin [- f J =>y = -f -1 t .". The principal value of sin (- 1) is - ~~ (v) Let cos - \ — j) = y, where < y < n, then -if f) 1 cos I — 2 J = .y=>cos;y = -2 cos;y = -cost => cos y = cos I n — ~l =>y = l ~T .". The principal value of cos - I - — I is 2; ia 3 -1 "• (vi) Let cosec (- 2) = y, where - ~~ < v < _l , „. „ f — 7t l - 7t cosec (-2)= y => cosec j = -2 = cosec I "T - ! ^>}) = -7~ -1 — ^ .". The principal value of cosec (-2) is —7- Example 6.59: (i) If cot - ( — 1 =9, find the value of cos9 (ii) If sin - ( —J = tan - >. find the value of x Solution: (i) cot -1 ( 7 J = => cot9 = - /. tanG = 7 7; _ u "" ^ LU _ 7 sec 9 = Vl+tan 2 9 = V*+49 237 sec 9 = 5^/2 => cos9 = ^ -l • -l f^\ n -1 n (11) tan x = sin lyl = Z •'• tan x = t n 1 1 Properties of principal inverse Trigonometric functions: Property (1): (i) sin - (sinx) = x (ii) cos - (cosx) = x (iii) tan - (tanx) = x (iv) cot - (cotx) = x (v) sec - (sec x)=x (vi) cosec - (cosec x)=x Proof: (i) Let sinx = y, then x = sin - (y) ... (1) .". x = sin - (sinx) by (1) Similarly, the other results may be proved. Property (2): (i) sin - I —J = cosec - x (ii) cos - I —J = sec - x (iii) tan - I —J = cot - x (iv) cosec - I —J = sin - x (v) sec - I —J = cos - x (vi) cot - (~) = tan - x x) ■ ' \x Proof: (i) Let sin 1 f A = 9 => sin9 = - ^> cosec9 = x ^> 9 = cosec - (x) ^> sin I — I = cosec x Similarly the other results can be proved. Property (3): (i) sin - (- x) = - sin - x (ii) cos - (- x) = n - cos - x (iii) tan - (- x) = - tan - x (iv) cosec - (- x) = - cosec - x (v) sec - (- x) = n - sec - x (vi) cot - (- x) = - cot - x 238 Proof: (i) Let sin - (- x) = 9 .'. - x = sin9 => x = - sin9 x = sin(- 9) => - 9 = sin" x => 9 = - sin" x => sin - (- x) = - sin" x (ii) Let cos" (- x) = 9 => -x = cos9 => x = - cos9 = cos (n - 9) => n - 9 = cos - x => 9 = n - cos x => cos - (- x) = n — cos - x Similarly the other results may be proved. Property (4): ... . -i -i n _i _i n _j _j 7i (l) sin x + cos x = y (n) tan x + cot x = y (m) sec x + cosec x = y Proof: (i) Let sin - x = 9 => x = sin9 = cos 1 -z— 9 => -l n Q cos x = y - 9 => -l n • -l cos x = 2 - sin x • -1 -It => sin x + cos x = 2 Similarly (ii) and (iii) can be proved. Property (5): If xy < 1, then tan -1 x + tan -1 y = tan -1 h ■*-] Proof: Let tan - x = 9i and tan - y = Q2 then tan9i = x and tan92 = y tan9i+tan92 x + y ^>tan(9i + 92) = ~, — 7 — q — 7 — 7T = ~, 1 z 1 - tan9l . tan92 I - xy 239 9j + 62 = tan 1 ( x + y 1 — xy. -1 -1 -1 ( x + y => tan x + tan y = tan Note: Similarly, tan - x- tan - y = tan - I 1 VL + xy. Property (6): sin Ix + sin 1 y = sin l \_x \jl-y +y\l-x \ Proof: Let 9i = sin - x and 62 = sin - y then sin9i = x and sin92 = y => sin(9i + 92) = sin9i cos92 + cos9i sin92 = (sinGi "ul - sin 92 + "\yl - sin 9l sin92) = UVTV + yVT^] => 9i + 9 2 = sin -1 [x-\ll-y 2 + y\jl-x 2 ] sin x + sin y = sin \_x\l-y +y V^ 2 ] Example 6.60: Prove that (i) tan I y J+tan I yr J = Solution: 11 1 _L tan -1± (ii)cos -11 _1 27 5 + tan 5 = tan yy -11. (i) tan \~\ + tan * 1 77J = tan 7 + 13 il i -, _i 4 4 3 (ii) Let cos 7=9 then cos9 = t .'. tan9 = 7 = tan M90J =tan -1 k = tan _1 4 _! 3 cos j = tan 7 _i4 _i3 _i3 _ x 3 _i .". cos 5+tan j = tan 7 + tan j = tan 4 + 5 _l27 y\ {3\i ~ tan 11 Example 6.61: Show that tan + tan y + tan z = tan A) v4 1 fx + y + z-xyz Solution: tan x + tan y + tan z =tan x + y 1 - xy. + tan z 1 - yz - zx - xy. 240 = tan = tan x + y I —xy + Z (x + y)z 1 — xy = tan X i-yi -z- xyz 1- -xy 1- -XV -xz -yz 1 — xy x + y + g - xyz 1 - xy - yz - zx_ —1 —1 ^ Example 6.62: Solve tan 2x + tan 3x = t _1_ _1_ IT _1 tan 2x + tan 3x = t => tan 2x + 3x l-6x 2 = tan" 1 (1) 5x l-6x z = 1 => 1 - 6x 2 = 5x .. 6x 2 + 5x - 1 = 1 i.e. (x + 1) (6x - 1) = => x = - 1 or t The negative value of x is rejected since it makes R.H.S. negative. .". x = t Example 6.63: Evaluate : (i) sinl cos - (tJJ (ii) cos I tan" jl (iii) sin lycos" t 3 3 Solution: (i) Let cos" t = 9 . Then, cos 6 = t :. sin cos ! -j =sin9 = Vl-cos 2 9 =-\/l _9_ 25 4 5 (ii) Let tan l f jj =9 then, tan9 = t .'. cos I tan" jl = cos9 = sec 9 Vl+tan 2 9 5 _i 4 4 (iii) Let cos t = 9 ; then cos 9 = t sin 1 _!4" 2 cos 5 Example 6.64: Evaluate : cos Solution: . 6 = sin" 1 /l-cosB 1 VTo : cos " . -l 3 . -i 5 " sin t + sin tt x ■ -l 3 . A 3 A 4 Let sin t = A .•. sinA = t => cosA = t 5 c i y Let sin" ~p? = B .". sinB = ~p? => cosB = tt 241 ". cos " ■ -l3 . -1 5" sin t + sin "pr = cos (A + B) = cosA cosB - sinA sinB fl 12 3 5) 33 _ V5 • 13 5 • 13j _ 65 EXERCISE 6.9 (1) Find the principal value of (i) sin - y (ii) cos - ( xj (iii) cosec - (- 1) (iv) sec -1 (- y{2) (v) tan -1 (a/3) (vi) cos -1 (- -j=) (2) Prove that (i) 2tan ( I = tan . (ii) 2tan x = sin 2 (iii) tan -1 (jj - tan -1 (jj = 4 (3) Evaluate: (i) cos( sin - tt I (ii)cos (4) Prove the following: -l (i) tan (iii) tan 1 - cosx 1 + cosx. 3 sin x : 2 (iii) tan I cos ~pyj (iv) 1 sin cos —1 2 —1 (ii) cos (2x - 1) = 2cos x 3x - X 1- 3jc 2 = 3tan x (iv) sin \2x "\j 1 - x ) = 2sin > (5) Prove that 2tan [ ~ = tan ! f -? 12 (6) Prove that tan (7) Solve : tan -1 'if x-l x- 2 -tan + tan" 1 l m — n \ _n rn + nj 4 \{x + 1 \ 71 _i / 2x \ _i f 1 — xf 1 1 IT (8) Solve tan | : , | + cot | ,-,,. J = T , where x > 1-x' x + 2 2 2x ■4 K 3 -1 -1 -1 4 (9) Solve : tan (x + 1) + tan (x - 1) = tan y (10) Prove the following: (i) cos -1 x + cos~ l y = cos -1 [xj - "\/ 1 - x 2 "\/ 1 - )> 2 ] (ii) sin -1 x - sin -1 ); = sin -1 \_x\jl -y -y\jl-x \ (iii) cos -1 x - cos -1 ;y = cos -1 [xy + V 1 -* 2 • V 1 ~y 2 J 242 OBJECTIVE TYPE QUESTIONS Choose the correct or most suitable answer (1) The order of matrix B = [1 2 5 7] is (l)lx 4 (2) 4 x 1 (3) 2 x 1 (4) 1 x 1 (2) Number of elements in a matrix of order 2 x 3 is (1) 5 (2) 2 (3) 3 (4) 6 (3) If A = "214" _-3 2 1. and X + A = then matrix X is (1) "214 .-3 2 1 (2) "_ 2 - 1 - 4~ 3 -2 -1. (3) ~-2 -1 -4 _ 3 2 1 (4) "2 1 4" _3 -2 -1. " 7" (4) The product of the matrices [7 5 31 3 ? is equal to (1) [70] (2) [49] (3) [15] (4) 70 "V2 o" (5) The type of the matrix ^3 V3. is (1) a scalar matrix (2) a diagonal matrix (3) a unit matrix rm (4) diagonal and scalar (6) if [: I y c -1] A' .3. = [i: 5] then the va ueo f a is (1)5 (2)2 (3) + 3 (4) ±4 (7) Matrix A is of order 2x3 and B is of order 3x2 then order of matrix B A is (1)3x3 (2)2x3 (3)2x2 (4)3x2 (8) If [3 -1 2]B = [5 6] the order of matrix Bis (1) 3 x 1 (2) 1 x 3 (3) 3 x 2 (4) 1 x 1 (9) The true statements of the following are (i) Every unit matrix is a scalar matrix but a scalar matrix need not be a unit matrix, (ii) Every scalar matrix is a diagonal matrix but a diagonal matrix need not be a scalar matrix. 243 (iii) Every diagonal matrix is a square matrix but a square matrix need not be a diagonal matrix. (1) (i), (ii), (iii) (2) (i) and (ii) (3) (ii) and (iii) (4) (iii) and (i) (10) The matrix "8 5 r 6 4 _0 2_ IS (1) the upper triangular (3) square matrix 2 -3 (11) The minor of 2 in (DO 6 is (2)1 2-3 5 6 4 1 5 -7 (1) - 18 (2)18 (12) The cof actor of - 7 in (2) lower triangular (4) null matrix is (3)2 (13) IfA = aj bi c l a 2 h c 2 «3 h c 3 (3) -7 and I A I = 2 then I 3 A I is (4) -3 (4)7 (1)54 (2)6 (3)27 (4) -54 (14) In a third order determinant the cofactor of a 2 3 is equal to the minor of a 2 3 then the value of the minor is (4)0 (4) x = (1)1 (15) The solution of (2)i 2x 3 2 3 = 0is (3) -A (1)JC=1 (2) ; 1 1 c = 2 1 (3) x = 3 (16) The value of 2x 2y lz is 3x 3y 3z (1)1 1 2 3 (2) > cyz 3 1 2 (3) x + y + z (17) IfA = 3 2 1 2 3 1 then 1 2 2 3 3 1 ie equal to (1)A (2)- -A (3)3A (4)0 (4) - 3A 244 (18) The value of the determinant 1 2 3 7 6 5 1 2 3 IS (1) (2) 5 (3) 10 (19) If A is a square matrix of order 3 then I kA I is (4) - 10 (l)fclAI (2) - k 1 A (3) r i a i (4)-*r IAI 1 4 3 2 8 6 (20) IfA = -11 5 3 2-1 and Aj = -2 6 2 10 4 -2 then (1) Aj = 2A (2) A l = 4A (3) A l = 8A (4)A = 8Aj 7 6 1 7 6 1 (21) IfA 1 = 5 3 8 and A 2 = 8 2 4 then 8 2 4 If 6 16 (l)Aj=-2A 2 (2)A 2 = -2Aj (3) Aj = 2A 2 (4)A,=-2A 2 (22) Two rows of a determinant A are identical when x = - a then the factor of A is (1) x + a (2) x — a (23) The factor of the determinant (3) (x + ay (4) (x - ay x -6 -1 2 -3i i-3 is - 3 2x x + 2 (1) x + 2 (2) x - 3 (3) 2jc + 1 (4) jc + 3 (24) If all the three rows are identical in a determinant A on putting x = a then the factor of A is (l)x-a (2)x + a (25) The factor of the determinant (l)jt (2)x + b (26) The value of the determinant (1) abc (2) o)( X - a y (4) (jc + a y x + a b c a x + b c is a b x + c (3)x + c (4)x-a + b + c a 2 b is c (3) a 2 b 2 c 2 (4) - abc 245 (27) The value of the product (2) -56 1 2 -3 1 (1)56 (28) IfA = 2 1 -4 (3)-l is a \ *1 C\ a 2 &2 c 2 «3 h c 3 (4) -63 and Aj, Bj, Cj are the cofactors of aj, ft 1; c l then a l A 2 + b l B 2 + c { C 2 is equal to (1)A (2)0 (3) -A (4) A 2 (29) Given that the value of a third order determinant is 1 1 then the value of the determinant formed by the respective co-factors as its elements will be (1)11 (2)121 (3)1331 (4)0 (l+ax) 2 (l+ay) 2 (1 + azf (2) 121 (30) A factor of the determinant (l)x + y (2)a + b (l+bx) 2 (l+by) 2 (1 + ftz) 2 (1 + ex) 2 (1 + cy) 2 (1 + cz) 2 is (3)jc-y (4) a + b + c -» -» -^> -^> -> -» (31) The position vector of A is 2 i + 3 j + 4 A; , AB = 5 i + 1 j + 6k then position vector of B is (1) 7 j + 10 v + 10 k -> -> -> (2) 7 j -10 j + 10 it (3)7? + 10 7 -10 it (4) -7 j +10 7 -10 A > (32) -> If a is a non zero vector and & is a scalar such that k a = 1 then Ac is equal to /1 \ -> 1 //I \ _i_ - i (1) a \L) i p; -> a (4)± -> a (33) -> -> >■ >■ Let a , b be the vectors AB , BC determined by two adjacent sides > of regular hexagon ABCDEF. The vector represented by EF is (1) a - ft (2) a + ft (3) : I a ;4>- ft 246 (34) If AB =k AC where k is a scalar then (1) A, B, C are collinear (2) A, B, C are coplanar (3) AB , AC have the same magnitude (4) A, B, C are coincident (35) The position vectors of A and B are a and b . P divides AB in the ratio 3 : 1 . Q is the mid point of AP. The position vector of Q is -> -> -» — » -» -> -» — » 5a + 3fr 3 a + 5 fc 5 a + 3& 3 a + fc (,-U g (A) 2 ^ ' 4 ^ ' 4 (36) If G is the centriod of a triangle ABC and O is any other point then OA + OB + OC is equal to (1) O (2) OG (3) 3 OG (4) 4 OG (37) If G is the centriod of a triangle ABC then GA + GB + GC is equal to / \ -* ~* -+ (1) 3 \a + b + c) (2) OG (3) O (4) 3 (38) If G is the centriod of a triangle ABC and G' is the centroid of triangle A' B' C ' then A A' + BB' + CC ' = (1) GG' (2)3GG' (3)2GG' (4)4GG' -» -» (39) If the initial point of vector - 2 i - 3 j is (- 1, 5, 8) then the terminal point is -> -» -> -» -> -» (1) 3 j + 2; + 8 * (2) - 3 j + 2 y + 8 k -> -> -> -» -> -> (3) - 3 i - 2 y - 8 * (4) 3 i + 2 y - 8 k (40) Which of the following vectors has the same direction as the vector i -2y (1)- i +2y (2)2/ +4y (3) - 3 » +67 4)3/ -67 247 (41) If a = j + _/' - 2 k , b = - i + 2 j + k , -» -» -» -» 2 fc , then a unit vector parallel to a + b + c is c = j - 2 y + '"^^^r^ 1 ^^ (4,^3^ (42) -> — » — » — » -» — » — » — » If a =2i +y - 8 it and fc = i + 3 j - 4 it -» -» of a + b = then the magnitude (1)13 (2)13/3 (3)3/13 (4)4/13 (43) If the position vectors of P and Q are -> -» -> -» -» -> 2i + 3 y -1 k , 4 i - 3 j + 4 k , then the direction cosines of PQ are 2 -6 11 -2 -6 -11 ( } VT61 ' VT61 ' VT61 ( } Vi6i 'VT6T 'VT6T (3)2,-6,11 (4)1,2,3 ax 2 3 (44) If (x + 2)(2x-3) = ^2 + 2^ thena = (1)4 (2)5 (3)7 (4)8 (45) If nPr = 720 nCr, then the value of r is (1)6 (2)5 (3)4 (4)7 (46) How many different arrangements can be made out of letters of words ENGINEERING 11! 11! 11! (Dll! (2) ^W (3) 3H2! < 4 >!T (47) The number of 4 digit numbers, that can be formed by the digits 3, 4, 5, 6, 7, 8, and no digit is being repeated, is (1)720 (2)840 (3)280 (4)560 (48) The number of diagonals that can be drawn by joining the vertices of an octagon is (1)28 (2)48 (3)20 (4)24 (49) A polygon has 44 diagonals then the number of its sides is (1)11 (2)7 (3)8 (4)12 248 (50) 20 persons are invited for a party. The number of ways in which they and the host can be seated at a circular table if two particular persons be seated on either side of the host is equal to (1)18! 2! (2) 18! 3! (3) 19! 2! (4) 20! 2! (51) If n is a positive integer then the number of terms in the expansion of (x + a) n is (1) n (2) n - 1 (3) n + 1 (4) n + 2 (52) The values of wCO - nC 1 + nC2 - nC3 +...(- 1)" . nCn is (1)2" +1 (2)n (3)2" (4)0 (53) The sum of the coefficients in the expansion of (1 - x) is (1) (2) 1 (3) 10 2 (4) 1024 24 (54) The largest coefficient in the expansion of (1 + x) is (1)24C24 (2)24C13 (3) 24C12 (4)24Cll r 2n 1 8 (55) The total number of terms in the expansion of i(a + b) \ is (1)11 (2)36 (3)37 (4)35 (56) Sum of the binomial coefficients is (1) In (2) n (3) 2" (4) n + 17 8 (57) The last term in the expansion of (2 + sj3) is (1)81 (2)27 (3)^/3 (4)3 (58) If a, b, c are in A.P., then 3", 3 b , 3 C axe in (1) A.P. (2) G.P. (3) H.P. (4) A.P. and G.P. (59) If the « th term of an A.P. is (In - 1) : then the sum of n terms is (1) n 2 - 1 (2) (2m - 1) (3)n 2 (4) n 2 + 1 (60) 2 The sum of n terms of an A.P. is n . Then its common difference is (1)2 (2) -2 (3) + 2 (4)1 (61) The sum to the first 25 terms of the : series 1 + 2 + 3 . . is (1) 305 (2) 325 (3) 315 (4) 335 (62) The n term of the series 3 + 7 + 13 + 21+31 + ... . is (1) An - 1 (2) n 2 + In (3) (n 2 + n + 1) (4) (n 3 + 2) (63) What number must be added to 5, 13 and 29 so that sum may form a G.P? (1) 2 (2) 3 (3)4 (4)5 249 (64) The third term of a G.P. is 5, the product of its first five terms is (1)25 (2)625 (3)3125 (4)625x25 (65) The first term of a G.P. is 1. The sum of third and fifth terms is 90. Find the common ratio of the G.P. (1)±2 (2)VlO (3) + 3 (4) -3 (66) When the terms of a G.P. are written in reverse order the progression formed is (l)A.P. (2) G.P. (3)H.P. (4) A.P. and H.P. (67) If A, G, H are respectively arithmetic mean, geometric mean and harmonic mean then (1)A>G>H (2)A<G>H (3)A<G<H (4) A > G < H (68) The A.M. between two numbers is 5 and the G.M. is 4. Then H.M. between them is (1)3 j (2)1 (3)9 (4)1 \ (69) If a, b, c are in A.P. as well as in G.P. then (1) a = b * c (2) a * b = c (3) a * b * c (4) a = b = c (70) The A.M., G.M. and H.M. between two positive numbers a and b are equal then (1) a = b (2) ab=\ (3)a>b (4)a<b 2 3 X X (71) e x = 1 + x + -j\ + 3T + is va lid for (1)-1<x<1 (2)- 1 <jc< 1 (3) all real x (4)jc>0 (72) <? log * is equal to (l)x (2)1 (3)e (4)log^ (73) The equation of x-axis is (1)jc = (2)x = 0,;y = (3)y = (4)jc = 4 (74) The slope of the straight line 2x - 3y + 1 = is -2 -3 2 3 (1)— (2)— (3)3 (4) 2 (75) The y - intercept of the straight line 3x + 2y - 1 = is (1)2 (2)3 {3)\ (A)-\ (76) Which of the following has the greatest j-intercept in magnitude? (l)2x + 3y = 4 (2)x + 2y = 3 (3)3x + 4y = 5 (A)Ax + 5y = 6 250 (77) If the equation of the straight line is y = -\J3 x + 4, then the angle made by the straight line with the positive direction of x-axis is (1)45° (2)30° (3)60° (4)90° (78) If the straight lines a^x + b^y + c^ = and a 2 x + b 2 y + c 2 = are perpendicular, then a V^=~V 2 ^7T 2 =V 2 V)a,a 2 = - bl b 2 (4)- =y 2 =- (79) Which of the following is a parallel line to 3x + Ay + 5 = 0? (1) 4jc + 3;y + 6 = (2)3jc-4j + 6 = (3) 4x - 3y + 9 = (4) 3x + Ay + 6 = (80) Which of the following is the equation of a straight line that is neither parallel nor perpendicular to the straight line given by x + y = (l)y = x (2)y-x + 2 = (3)2y = 4x+l (4)y+x + 2 = (81) The equation of the straight line containing the point (-2, 1) and parallel to 4x - 2y = 3 is (l)y = 2x + 5 (2)y = 2x-l (3)y = x-2 (4)y = ^x (82) Equation of two parallel straight lines differ by (l)xterm (2) y term (3) constant term (4)xyterm 2 (83) If the slope of a straight line is t , then the slope of the line perpendicular to it, is 2 2 3 3 (1) 3 (2) -J (3) 2 (4) -J (84) The graph of xy = is (1) a point (2) aline (3) a pair of intersecting lines (4) a pair of parallel lines 2 2 (85) If the pair of straight lines given by ax + 2hxy + by = are perpendicular then (l)ab = (2)a + b = (3)a-b = (4)a = 251 2 (86) When h =ab the angle between pair of straight lines 2 2 ax + 2hxy + by = is (l)f (2)f (3)f (4)0° 2 2 (87) If 2x + 3yx - cy = represents a pair of perpendicular lines then c = (l)-2 (2) -| (3)2 (4) | 2 2 (88) If 2x + kxy + Ay =0 represents a pair of parallel lines then k = (1)±32 (2) ±2^2 (3) +4 a/2 (4) ± 8 2 2 (89) The condition for ax + 2hxy + by + 2gx + 2fy + c = to represent a pair of straight lines is (\)abc + 2fgh -bf- ag 2 -ch 2 = (2) abc - 2fgh - ag 2 - bf - ch 2 =0 (3) abc + 2fgh - ah 2 -bg 2 -cf = (4) abc + 2fgh - af 2 - bg 2 - ch 2 =0 (90) The length of the diameter of a circle with centre (2, 1) and passing through the point (-2, 1) is (1)4 (2)8 (3)4^/5 (4)2 2 2 (91) Given that (1, - 1) is the centre of the circle x +y + ax + by - 9 = 0. Its radius is (1)3 (2) >/2 (3) VII (4)11 (92) The equation of a circle with centre (0, 0) and passing through the point (5,0) is (l)x 2 + y 2 -10x = (2)x 2 + y 2 = 25 (3) x 2 + y 2 + 10* = (4)x 2 + y 2 -10y = 2 2 (93) The radius of the circle x +y -2x + 4j-4 = 0is (1)1 (2)2 (3)3 (4)4 2 2 (94) The centre of the circle x +y +2x-4j-4 = 0is (1)(2,4) (2) (1,2) (3) (-1,2) (4) (-2, -4) 252 (95) If 2x + 3y = and 3x - 2y = are the equations of two diameters of a circle, then its centre is (l)(l,-2) (2) (2, 3) (3) (0,0) (4) (-3, 2) 2 2 (96) If the line y = 2x - c is a tangent to the circle x + y =5, then the value of c is (1)±5 (2)+V5 (3) + 5a/5 (4)±5-y/2 2 2 (97) The length of the tangent from (4, 5) to the circle x +y = 25 is (1)5 (2)4 (3)25 (4)16 (98) If the circle has both x and y axes as tangents and has radius 1 unit then the equation of the circle is (l)x 2 + (y-l) 2 =l (2)x 2 + y 2 =l (3)(x-l) 2 + (y-l) 2 =l (4)(x-l) 2 + /=l 2 2 (99) Which of the following point lies inside the circle x + y - 4x+2y - 5=0 (1)(5, 10) (2) (-5, 7) (3) (9,0) (4) (1, 1) (100) The number of tangents that can be drawn from a point to the circle is (1)1 (2)2 (3)3 (4)4 (101) If two circles touch each other externally then the distance between their centres is (102) The number of points in which two circles touch each other internally is (1)1 (2)2 (3)0 (4)3 (103) One radian is equal to (interms of degree) 180° ,„ n ,„ 180 _ 11 (D-rr- (2)— (3)— (4) 11 w 180° v; n ^'180° (104) An angle between 0° and - 90° has its terminal side in (1)1 quadrant (2) III quadrant (3) IV quadrant (4) II quadrant 253 (105) tTq of a complete rotation clockwise is (1)-1° (2) -360° (3) -90° (4)1° (106) If the terminal side is collinear with the initial side in the opposite direction then the angle included is (1) 0° (2) 90° (3) 180° (4) 270° (107) Area of triangle ABC is (1)2 ab cosC (2) 2 ab sinC (3) y ab cosC (4) y be sinB (108) The product s(s - a) (s - b) (s - c) is equal to A (1)A (2) A 2 (3)2A (4)^ (109) In any triangle ABC, A is abc abc abc (I) abc (2) W (3)2R (4)-r- (110) In triangle ABC, the value of sinA sinB sinC is A ... A ... A A — 2 ( 4 ) — 2R 2 4R d>2R ( 2 >4R 0)^2 (4) AP 2 (111) cosB is equal to c 1 + a 2 - b 2 ^ (1) lea {V> 2bc (3) lab (4) lab c 2 + a 2 - b 2 ... c 2 + b 2 - a 2 ... a 2 + b 2 - c 2 . .. a 2 + b 2 + c 2 254 (1) (i) ■2 3 4' 3 4 5 4 5 6. (ii) ANSWERS EXERCISE 1.1 1 2 3" 2 4 6 3 6 9. (2) x = 0, j = 7, z = 3 (4) (i) (v) (6) X = ^ 3 5 1 -2-7 -12 4 4 -4 -6 10. (ii) _ 2 -12. (iii) _6 4 "6 -f "6 -1 - "9 " _5 2 _ (vi) _5 2 (vii) .14 -16. "22 -1" "-2 -5 -1" -2 3 -3 ,Y = -11 3 _-2 1 -3_ _ -7 2 (iv) (viii) 4 -4 6 4 . 8 ' 18 -7. (8) k = 1 (10) jc=1,-3 (ll)x = 2,-5 (13) | EXERCISE 1.2 9 3' -6 7. (14)jc=l,y = 4 27 (1)0 (2) (i) non-singular (ii) singular (3) (i)x = ~3~ (ii)x = 9 (4) (i)0 (ii) (6)a 3 + 3a 2 EXERCISE 1.3 (3) x = 0, 0, - (a + b + c) (4) (a - b) (b - c) (c - a) (ab + be + ca) EXERCISE 2.1 > -> -> > -> -> (1) AC = a + b , BD = b - a EXERCISE 2.2 (1) 5~t +5~f +5~k , 5^/3 (2) VI 85 255 > -> -> -> ► -> -> -> (3) AB = - 3 i - j -5 k ; BC = 4 j - 7 j + 7 jfe ; >• -> -> -> CA =- i +8 j -2* AB = ^35 , BC = VTTi , CA = ^69 m± 17? -A- lot «»2(t + ? + t) — >• -> -> -> M -5 iO (13) pq =4, -5; + n* ;^^,^J (16) non-coplanar vectors. EXERCISE 3.1 1 _ 1 20 _ _D_ (1) 2(jc-1) 2(x+l) {2 'x-3 x-2 3 _ _7_ 13 ( ' 2(x - 1) "x-2 + 2(jc - 3) ( 4 ) 9fr7T) " 9(^T2) " ^7^2 _ -4 4 1 ^ 9(* + 2) + 9(x-l) 3 (x-l) 2 2 3 2 (6) 25(x^2) + i^T^2 "25(xT3) ^7 J_ 9 (7) 2x + x 2 + 2(x + 2) 2 3 9 (8) 7T-^ + (x - 2) ( x _ 2) 2 (jc - 2) J , Q , 1 4x-8 ^ 5(x + 2) + 5(x 2 +1) 256 (10) 1 (x ~ 3) ^ 1U )2(x + l) 2(x 2 +1) x-5 4 A' - (11) ^l^T + 3.-2 (12) 1-777+-^ EXERCISE 3.2 (1) 378 (2) 42 (3) 600 (4) 1320 (5) 42840 (6)512 (7)153 (8) (i) 27216 (ii) 90000 (9)5x5! (10)21 (11) 2 5 (12)9000 (13) (i) 125 (ii) 60 (14) 2 5 EXERCISE 3.3 (1) (i)60 (ii) 2730 (iii) 120 (iv)^f (v) 15120 (2) 23 (3)4 (4)41 (7)172800 (8)5040 (9)60 (10)93324 (11)34650 (12) (i) 840 (ii) 20 (13) 9000 (14) 4 5 (15) (i) 8! (ii) 7! (16) ^r EXERCISE 3.4 (1) (i) 45 (ii) 4950 (iii) 1 (2) 23 (3) 3 (4) 45 (5) (i) 12 (ii) 8 (6) 19 (7) 7 EXERCISE 3.5 (1) 66 (2)200 (3)210 (4)425 (5) (i) 15 C n (ii) 14 C 10 (iii) i 4 C n (6)780 (7) (i)40 (ii) 116 (8)1540 (9)817190 EXERCISE 3.7 (1) (i) 243a 5 + 2025 A + 6750a V + 1 1250a V + 9375a6 4 + 3125 b 5 (ii) a 5 - 10 A + 40a V - 80a V + Wab 4 - 32b 5 (iii) 32jc 5 - 240jc 6 + 720jc 7 - 1080jc 8 + 810x 9 - 243.x 10 257 ,. , 11 llx 10 55x 9 165x 8 330x 7 462x 6 462x 5 330x 4 (IV) x + — — + —j- + —3- + —4- + — ^ + — g- + ~y~ J j y y y y y 165x 3 55x 2 llx J_ + / + / V° V 1 (v) x 12 + 12x 10 y 3 + 60x 8 / + 16Q*V + 240x 4 ;y 12 + 192*V 5 + 64j 18 ,, -n 4 2 , . 7/2 3/2 , , 3 3 , A 5/2 7/2 , 2 4 (vi) x>>+4x j + 6x j + 4x y + x y (2) (i) 58 ^2 (ii) 152 (iii) 352 (iv) 128a 3 + 4320a 2 + 9720a + 1458 (v) 5822 ^3 (3) (i) 1030301 (ii) 970299 (4) 0.9940 4 , 2 !6C8 -a 8 (5) (i) 8C4 2V 2 (ii) 16C 8 (iii) j — x (iv) 13C 6 .2 6 x 7 / and - 13C 7 . 2 7 x 6 / (v) 17C 8 .2 8 y and 17C 9 .2 9 yj x x (7) - 165 (8) (i) 7920 (ii) 2268 (iii) i 2 C 4 [-J 9 8 (9) r=3 (10) 7, 14 EXERCISE 4.1 (1) (i) 25, - 125, 625, - 3125 and 15625 (ii) § , | , y , 21, y (iii)- 1,-12, -23, -34, -45 (iv) | , f , |, |, f ..222 1 4 1 16 25 (v)j , 0, 3 , 0, 3 (vi) 3 , 9 , 3 , gj-, 243 (2) (i)y , y (ii) 1,0 (iii)y , yf (iv) 64, -512 5 17 37 (3) 0, y 8, y, 24, y 258 (4) (i) 2, 2, 1, 0, -1 (iii) 1,2,6,24, 120 (ii) 1, 2, 3, 5, 8 (iv) 1,1,5, 13,41 (5) 1 1 3" (6)4 [5 n -l] EXERCISE 4.2 (1) (i) 4, 7, 10, 13, 16 (ii) 5, 7, 9, 11, 13, 15 (7)doo(2 10 °-D (4) 19 (2) (i) 10 (ii) 1 (m)P (6) 6, 24 (10) 2, 3, 6 (or) 6, 3, 2 EXERCISE 4.3 CD ©^ x ; + ■ 40 160 "+... (ii) 1 >/6 2 -3 £ X_ IX 1+ 6 + 18 + 324 + H-9.7.5 12 (2) (i) 10.01 (ii) 0.2 (5)^ ... (r+1) (r+2) (r+3) r ® L23 x EXERCISE 5.1 (1) x 2 + /-2jc + 8)'-19 = (2)3x + j = 2 (3) (i)f=l (ii)P(l,2) (4)/-24x 2 = (7) (i) x 2 + y 2 + x - 3y + 2 = (ii) 15x 2 + I5y 2 + 66x - 96y + 207 = EXERCISE 5.2 (1) 4jc-7y-10 = (2)y = 3x + 4 (3)x-y = 6 (4) llx-y = 27 (5)2x + y = 6;x + 2y = 6 (6)x + 3y = 8 (7) 3x - 2y = ; 2x - y = and 5x - 3y = 14 (8)-jr^ units VT3 (9) 2jc-3)'+12 = (10)9jc-8)'+10 = 0;2jc-j = 6 (1 1) 2x - 3y = 6; 3x - 2y = 6 (12) x intercept ~ ; y intercept 2 (13) (8, 0) and (- 2, 0) (14) 3^2 units 259 EXERCISE 5.3 (1) f (3)3;c + 2y+l=0 (4)jc-y-l=0 (5) (1, 2) (6) k = -9 (7) 2 mits (8)p=l;p = 2 (9) 28x + ly - 74 = (10) 5x + 3y + 8 = (ll)x + y=l (12) 5x + 3y + 5 = (14)J;| (17) a = 5 (18)a = Tf d9) (2, f) (21) d, 12) (22) (- 4, - 3) EXERCISE 5.4 (1) a = 2 ; c = - 3 (2) nl3 (4) 1 (6) 2jc 2 - 3xy -2y 2 = (7) 3x 2 + 7xy + 2/ - 4x + ly - 15 = (8) k = - 1 ; Ax - 3y + 1=0 and 3x + 4y- 1=0 ; nil (9) C = 2 ; 6jc - 2y + 1 = and 2x -y + 2 = ; tan" 1 (1/7) (10) £ = -10 ; 3jc-2j+l=0and4x + 5)' + 3 = EXERCISE 5.5 (1) (i) (0, 0) ; 1 (ii) (2, 3) ; a/22 (iii) (4, 3) ; 7 f 2 T\ 2V5 ,— (iv) r3'3"J ; 3 (v)(4,4);VT0 (2) a = 4 ; b = 2 ; 2x 2 + 2y 2 + Ax + Ay - 1 = (3) x 2 + y 2 -Ax-6y+U=0 (A) x 2 + y 2 -6x -Ay -12 = (5) jc 2 + /-14jc + 6)' + 42 = (6) x 2 + y 2 + 8x-l0y + 25 = (7) 2-\jlb n unit ; 10^ square units (8) x 2 + y 2 - Y2x + 11 = ; x 2 + y 2 + Ax - 21 = (9) x 2 + y 2 -3x-6y+l0 = 260 (10) x 2 + y 2 =l (11) x 2 + y 2 -5x-y + 4 = (12) x 2 + y 2 -6x-8y+l5 = (13) x 2 + y 2 -4x-2y-5 = (14) 16x 2 + I6y 2 = 1 3 3 (15) x = 2 cos9;>' = 2' sin 9 EXERCISE 5.6 (1) 2^5 units (3) y - 1 = (4) outside (5) (0, 0) and (4, - 3) lies inside; (- 2, 1) lies outside (6) (0, 2) ; (2, 0) (7) 2x + y = ± 3^5 (8) 5a/2 units (9) (i)x 2 + ;y 2 -10jt:-12;y + 25 = (ii) x 2 + y 2 - 10* - 12y + 36 = (10) 4x + 3y + 6 = (12) x-5y+19 = (ll)(i) x + y = ±4>/2 (u)x-y = ±4^[2 (13) + 40 EXERCISE 5.7 , 1 5 (14)1-4-4 (3) J + f-lx -6^-39 = (4) x 2 + y 2 -8x +12^-49 = (6) (i) x 2 + y 2 - 2x + 2y + 1 = (ii) x 2 + y 2 - 6x - 4y - 44 = (7) x 2 + y 2 - \6x - 18y - 4 = (8) 3x 2 + 3j 2 - 14jc + 23y - 15 = EXERCISE 6.1 (1) (i)t (n)"5- (m) -5— (iv)— 5— (v)— ^7— (vi) g W 9 v"V 9 Viv; 9 vv; jg (2) (i) 22° 30' (ii) 648° (iii) - 171°48' (app.) (iv) 105° (3) (i) Qj (ii) Q 3 (iii) Qi EXERCISE 6.2 1331 24 (1) 276 (2) (i) - sin 60° (ii) - cos40° (iii) tanl0° (iv) - tan60° (v)cosec60° (vi)-sin30° (vii) cos 30° 261 (5) (i) - cosec A (ii) - sec A (iii) - cotA (iv) cosA (v) tan A 1 (8) (i) V2 (v)-l (vi) a/3 .... \/3 (")" 2 (vii) 1 (iii) - a/2 1 (iv)- 2 (viii) V3 (10) (i)y (ii) 1 (iii)0 (iv r 2 4 ^ 6 (v)5 (vi)2 (vii)- 25 (viii) 1 1 (ix) J2 (1) (i)^ (ii) w 120 EXERCISE 6.4 \/6-a/2 (iii) 2 + V3 (iv)^^ , A a/2+1 a/6 + A /2 x/2-^/3 a/6 + A /2 (8) vv 2 (12) a/2 + 1 (3) i (14)^^ (6) (i) ^ (1) (i) sin69 + sin29 (iv) cos2A - cos4 A 1 EXERCISE 6.5 ^125 EXERCISE 6.6 (ii) cos 149 + cos29 (v) sin9A - sin3A 1 (iii) sinlOO - sin49 (vi>2 [sinl39 + sin59] 1 (vii) ^ [sin2A - sin A] (viii) y [sin6A + sin A] (ix) y [cos39+cos 9/3] (2) (i) 2sin9A cos4A (ii) 2 cos9A sin4A (iii) 2cos9A cos4A (v) 2 cos42° sin 10° (viii) 2 sin30° cos20 c (iv) - 2sin 9A sin4A (vii) 2 cos50° sin30° (vi) 2 cos37° cos 14° (ix)2sin30°cosl0° 53° 17° (x) 2 cos y cos —j- (1) (i)" (ll) T (ill) T EXERCISE 6.7 (iv)* (v) (vi) n n 7. (vn) 7 262 ,„. ,.. W7T , ,,„ It ,.. N 71 ,... N 2n7t 7t (2) (i) y + (- 1)" 12 W "" " 3 ( m ) "3" ± 4 71 ?27E 71 (3) (i) tin ± 7 , Wi (ii) y , (2n +1)2 (iii) "Tt 71 271 71 ?17l (4) (i) 2htt ± T (ii) 2«7t + 71 (iii) 2«7t + y , (In + 1) T (iv) y , 2«7t (5) (i) 2mtt + 4 (or) nn + (- l) n ^ - 4 7t 7C 271 (ii) 2«7t - v (iii) 2«7t - t (iv) 2«7t, 2rai + y EXERCISE 6.9 u u n 3tt k 3k (1) (1)3 (11)3 (111)- 2 (iv)y (v)j (vi)y (3) (i){§ (ii)| (iii)y (iv)y (7)* = ±n| (8) 2-V3 (9)* = ^ 263 Objective Type Questions - Answers (Key) (1) 1 (2)4 (3)2 (4)1 (5)2 (6)4 (7)1 (8)3 (9)1 (10)1 (11)1 (12)2 (13)1 (14)4 (15)1 (16)4 (17)2 (18)1 (19)3 (20)3 (21)2 (22)1 (23)4 (24)3 (25)1 (26)3 (27)2 (28)2 (29)2 (30)3 (31)1 (32)4 (33)4 (34)1 (35)1 (36)3 (37)3 (38)2 (39)2 (40)4 (41)4 (42)1 (43)1 (44)3 (45)1 (46)2 (47)1 (48)3 (49)1 (50)1 (51)3 (52)4 (53)1 (54)3 (55)3 (56)3 (57)1 (58)2 (59)3 (60)1 (61)2 (62)3 (63)2 (64)3 (65)3 (66)2 (67)1 (68)1 (69)4 (70)1 (71)3 (72)1 (73)3 (74)3 (75)3 (76)2 (77)3 (78)3 (79)4 (80)3 (81)1 (82)3 (83)4 (84)3 (85)2 (86)4 (87)3 (88)3 (89)4 (90)2 (91)3 (92)1 (93)3 (94)3 (95)3 (96)1 (97)2 (98)3 (99)4 (100) 2 (101)4 (102) 1 (103) 3 (104) 4 (105) 1 (106) 3 (107) 2 (108)2 (109) 2 (110)3 (111)1 264